Computation of Wiener and Hyper Wiener Indices in Several B-Complex Vitamins

Computation of Wiener and Hyper Wiener Indices in Several B-Complex Vitamins | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Computation of Wiener and Hyper Wiener Indices in Several B-Complex Vitamins T. Greeta, G. Jayalalitha This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4402009/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract B vitamins are crucial for the proper functioning of our body. Vitamin B complex, comprising eight essential vitamins, supports growth and sustains energy levels. Graph theory aids chemists in systematically molecular modeling structures and reactions. Topological indices serve as mathematical descriptors, analyzing physicochemical properties. In this work, four B complex Vitamins—B1, B2, B3, and B6—are examined. The investigation centers on utilizing the distance matrix. It computes both the Wiener Index and Hyper Wiener Index, providing valuable perceptions toward the structural attributes of these vitamins. The outcomes of these metrics are systematically compared with the respective molecular weights of the vitamins. This comparison offers a comprehensive understanding of their chemical characteristics. It delves into the relationship among the structural features, topological indices, and molecular weights of essential B-complex vitamins. This paper explores enhancing our understanding of their biochemical roles and potential applications in health and medicine. Chemical graph theory Vitamin B Complex Distance matrix Dominating number Topological indices Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Introduction In a graph G=(V,E), the vertices in V represent the collection of points denoting atoms, and E signifies the cluster of connections (bonds) that illustrate the linking of atoms within the molecule[ 1 ]. The significant contributions made in the domain of chemical graph theory have led to recent innovations and developments in chemical graph simulation and mathematical modelling. Chemical graph theory is crucial for understanding the structure of chemical compounds. Scientific studies consistently show that the properties of chemicals and medications are closely tied to their molecular makeup. In this theory, atoms are like points (vertices), and bonds are like connections (edges) between these points [ 2 ]. Topological indices, numerical representations of the structural aspects of chemical substances, are crucial in chemical graph theory. These indices, often referred to as graph invariant indices, provide quantitative insights into the topology of chemical structures. They serve a vital contribution in extracting quantitative structure-property or structure-activity relationships (QSPR/QSAR) [ 3 , 4 , 5 ]. These indices can be used independently or in conjunction with other quantitative variables to model and predict various properties or activities of chemical compounds. Chemical graph theory finds extensive implementations in pharmaceutical innovation, contributing significantly to the understanding and advancement of these fields. Distance-based topological indices are mathematical measures obtained from the structural graph of a chemical substance [ 6 ]. These indices capture important structural information by considering the distances between pairs of particles within the compounds. These indices play a crucial role in quantitative structure-activity relationship (QSAR) studies, cheminformatics, and computational chemistry. In the realm of graph theory, the dominating number focuses on vertex dominance. A dominating set in a graph comprises a subset of vertices, where each vertex either belongs to the dominating set or is adjacent to a vertex in the dominating set. [ 7 ]. γ(G), the dominating number, indicates the size, i.e., the number of vertices, in the smallest dominating set within a given graph. Alternatively, graph eccentricity evaluates the longest path between two vertices, also known as the greatest distance or furthest separation between any pair of vertices in the graph. [ 8 ]. Raji and Jayalalitha [ 9 ] conducted a study in which they explored the hyper Wiener index of the molecular graph of naphthalene, employing a domination-based approach. The Vitamin B complex, consisting of a group of water-soluble vitamins, is essential for numerous physiological functions such as energy metabolism, nerve function, and cell division. B vitamins, when combined with other micronutrients, are vital for maintaining overall health and well-being. These vitamins significantly influence energy levels, cognitive function, and cellular metabolism [ 10 ]. Comprising eight distinct vitamins, this complex is renowned for its diverse and vital contributions to human health. In this paper, our attention is directed toward the initial quartet of B vitamins—B1, B2, B3, and B6. The aim of this study is to compute topological indices for the four B-vitamins. The investigation includes the exploration of various theorems derived from these indices. Ultimately, a graphical representation is presented to illustrate the correlation between the computed indices and the physical properties of the vitamins. In Section 2 , the materials and methods used for the analysis are detailed. Section 2.1 focuses on calculating the Wiener and hyper-Wiener indices for vitamin B1 using the Distance Matrix method. Section 2.2 describes the calculation of these indices for vitamin B2, also using the Distance Matrix method. Section 2.3 deals with the calculation for vitamin B3, again utilizing the Distance Matrix. Section 2.4 explains the method for calculating the Wiener and hyper-Wiener indices of vitamin B6 using the Distance Matrix. The proof for Theorem 2.5 and Proposition 2.6 is presented. Moving to Section 3 , the results of the analysis are discussed. Section 3.1 specifically explores the correlations between the physical properties of the vitamins and their respective topological indices. Materials and Methods Simple graphs are used as models for the representation of B-vitamins. Certainly, the four B-complex vitamins under consideration for analysis are Vitamin B1 (Thiamine), Vitamin B2 (Riboflavin), Vitamin B3 (Niacin), and Vitamin B6 (Pyridoxine). Main findings revolve around employing distance-based topological indices for several B-vitamins. The chemical structures of Vitamin B1, B2, B3, and B6 were obtained from online sources. In our investigation, we specifically concentrated on molecular graphs of these compounds where hydrogen atoms were omitted[ 2 ]. This choice was deliberate as vertices representing hydrogen atoms were excluded due to their negligible contribution to graph isomorphism. The Wiener index stands as the inaugural generation of topological indices, a concept pioneered by the chemist Harry Wiener in 1947[ 11 , 12 ]. In his seminal work, Wiener introduced the term path number for graphs, representing the total of distances between pairs of carbon atoms within molecules. This innovative concept marked the inception of quantitative measures to capture the molecular structure's inherent connectivity patterns, laying the foundation for subsequent advancements in the field of chemical graph theory. In their paper titled " On Conjecture of Merrified Simmons Index ," Suresh Elumalai and colleagues provided a clear explanation of this matter[ 13 ]. The Wiener Index is defined as half of the total of distances between every pair of atoms in a molecular graph [ 14 ]. Mathematically, for a graph G with n vertices (atoms) $$W\left(G\right)=\frac{1}{2}\sum _{i=1}^{n}\sum _{j=1}^{n}{d}_{ij}$$ 1 Here, \({d}_{ij}\) denotes the minimum path length between vertices i and j within the molecular graph. The Wiener index is often used in cheminformatics and quantitative structure-activity relationship (QSAR) studies [ 15 ] to predict molecular properties based on molecular structure. The Hyper Wiener Index is an extension of the Wiener index and was introduced to incorporate additional structural information [ 16 ]. It is defined as $$WW\left(G\right)=\frac{1}{2}\sum _{i=1}^{n}\sum _{j=1}^{n}{\left[d\right({v}_{i},{v}_{j})}^{2}+d\left({v}_{i},{v}_{j}\right)]$$ 2 The Hyper Wiener index provides a more detailed measure of connectivity patterns within a molecule by considering both the distance from atom i to atom j and the distance from atom j to atom i. It enhances the representation of molecular structure and has applications in quantitative structure-activity relationship studies. When transforming the molecular structure into a molecular graph, atoms are translated into vertices, and the bonds between atoms are represented as edges [ 1 ]. Let G 1 denote the molecular graph of Vitamin B1, G 2 represent the molecular graph of Vitamin B2, G 3 symbolize the molecular graph of Vitamin B3, and G 4 stand for the molecular graph of Vitamin B6. 2.1 Wiener and Hyper Wiener Indices for Vitamin B1 Thiamine, or vitamin B1, is a water-soluble vitamin that is a member of the B-complex family. As a cofactor for enzymes involved in the transformation of carbohydrates into energy, it is essential to energy metabolism. The chemical formula for Vitamin B1 is C12H17N4OS. Figure 1 represents the chemical structure of Vitamin B1, and Fig. 2 depicts its molecular graph, denoted as Graph G1. Graph G1 has 18 vertices and 19 edges. The distance matrix for vitamin B1 is \(\begin{array}{ccc}{ \varvec{v}}_{1 }& {\varvec{v}}_{2} & \begin{array}{ccc}{ \varvec{v}}_{3}& {\varvec{v}}_{4}& \begin{array}{ccc} {\varvec{v}}_{5}& { \varvec{v}}_{6}& \begin{array}{ccc}{ \varvec{v}}_{7 }& {\varvec{v}}_{8}& \begin{array}{ccc}{ \varvec{v}}_{9 }& {\varvec{v}}_{10 } & \begin{array}{ccc}{\varvec{v}}_{11}& { \varvec{v}}_{12 }& \begin{array}{ccc}{\varvec{v}}_{13 }& {\varvec{v}}_{14 }& \begin{array}{ccc}{\varvec{v}}_{15 }& {\varvec{v}}_{16}& \begin{array}{cc}{ \varvec{v}}_{17}& {\varvec{v}}_{18}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\) \(\begin{array}{c}{\varvec{v}}_{1}\\ {\varvec{v}}_{2}\\ \begin{array}{c}{\varvec{v}}_{3}\\ {\varvec{v}}_{4}\\ \begin{array}{c}{\varvec{v}}_{5}\\ {\varvec{v}}_{6}\\ \begin{array}{c}{\varvec{v}}_{7}\\ {\varvec{v}}_{8}\\ \begin{array}{c}{\varvec{v}}_{9}\\ {\varvec{v}}_{10}\\ \begin{array}{c}{\varvec{v}}_{11}\\ {\varvec{v}}_{12}\\ \begin{array}{c}{\varvec{v}}_{13}\\ {\varvec{v}}_{14}\\ \begin{array}{c}{\varvec{v}}_{15}\\ {\varvec{v}}_{16}\\ \begin{array}{c}{\varvec{v}}_{17}\\ {\varvec{v}}_{18}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\left[\begin{array}{cccccccccccccccccc}0& 1& 2& 3& 4& 4& 3& 2& 3& 4& 5& 6& 6& 6& 5& 7& 8& 9\\ 1& 0& 1& 2& 3& 3& 2& 1& 2& 3& 4& 5& 5& 5& 4& 6& 7& 8\\ 2& 1& 0& 1& 2& 2& 3& 2& 3& 4& 5& 6& 6& 6& 5& 7& 8& 9\\ 3& 2& 1& 0& 1& 1& 2& 3& 4& 5& 6& 7& 7& 7& 6& 8& 9& 10\\ 4& 3& 2& 1& 0& 2& 3& 4& 5& 6& 7& 8& 8& 8& 7& 9& 10& 11\\ 4& 3& 2& 1& 2& 0& 1& 2& 3& 4& 5& 6& 6& 6& 5& 7& 8& 9\\ 3& 2& 3& 2& 3& 1& 0& 1& 2& 3& 4& 5& 5& 5& 4& 6& 7& 8\\ 2& 1& 2& 3& 4& 2& 1& 0& 1& 2& 3& 4& 4& 4& 3& 5& 6& 7\\ 3& 2& 3& 4& 5& 3& 2& 1& 0& 1& 2& 3& 3& 3& 2& 4& 5& 6\\ 4& 3& 4& 5& 6& 4& 3& 2& 1& 0& 1& 2& 2& 2& 1& 3& 4& 5\\ 5& 4& 5& 6& 7& 5& 4& 3& 2& 1& 0& 1& 1& 2& 2& 2& 3& 4\\ 6& 5& 6& 7& 8& 6& 5& 4& 3& 2& 1& 0& 2& 3& 3& 3& 4& 5\\ 6& 5& 6& 7& 8& 6& 5& 4& 3& 2& 1& 2& 0& 1& 2& 1& 2& 3\\ 6& 5& 6& 7& 8& 6& 5& 4& 3& 2& 2& 3& 1& 0& 1& 2& 3& 4\\ 5& 4& 5& 6& 7& 5& 4& 3& 2& 1& 2& 3& 2& 1& 0& 3& 4& 5\\ 7& 6& 7& 8& 9& 7& 6& 5& 4& 3& 2& 3& 1& 2& 3& 0& 1& 2\\ 8& 7& 8& 9& 10& 8& 7& 6& 5& 4& 3& 4& 2& 3& 4& 1& 0& 1\\ 9& 8& 9& 10& 11& 9& 8& 7& 6& 5& 4& 5& 3& 4& 5& 2& 1& 0\end{array}\right]\) The Wiener index of Vitamin B 1 is calculated as (by using Eq. ( 1 )) \(W\left({G}_{1}\right)=\frac{1286}{2}\) = 643 (3) The Hyper Wiener index of vitamin B 1 is calculated as (by using Eq. ( 2 )) $$WW\left({G}_{1}\right)=\frac{1}{2}\left(7120+1286\right)=4202$$ 4 2.2 Wiener and Hyper Wiener Indices for Vitamin B2 Vitamin B2, or Riboflavin, is an essential water-soluble vitamin that is a part of the B-complex family. It helps turn food into energy by being an essential component of several metabolic processes. Riboflavin is essential for the maintenance of healthy skin, eyes, and nerve functions. The chemical formula for Vitamin B2 is C 17 H 20 N 4 O 6 . Figure 3 represents the molecular structure of Vitamin B2. Figure 4 illustrates the molecular graph of Vitamin B2, denoted as G2. Graph G2 consists of 27 vertices and 29 edges. The distance matrix of Vitamin B2 is \(\begin{array}{ccc}{ \varvec{v}}_{1 } & {\varvec{v}}_{2}& \begin{array}{ccc}{ \varvec{v}}_{3}& {\varvec{v}}_{4}& \begin{array}{ccc}{\varvec{v}}_{5} & {\varvec{v}}_{6}& \begin{array}{ccc}{ \varvec{v}}_{7}& {\varvec{v}}_{8} & \begin{array}{ccc}{\varvec{v}}_{9}& { \varvec{v}}_{10 }& \begin{array}{ccc}{\varvec{v}}_{11 }& {\varvec{v}}_{12}& \begin{array}{cc}{\varvec{v}}_{13 }& \begin{array}{ccc}\cdots & \cdots & \begin{array}{cc}{ \varvec{v}}_{26 }& {\varvec{v}}_{27}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\) \(\begin{array}{c}\begin{array}{c}{ \varvec{v}}_{1 }\\ {\varvec{v}}_{2}\end{array}\\ {\varvec{v}}_{3}\\ \begin{array}{c}{\varvec{v}}_{4}\\ {\varvec{v}}_{5}\\ \begin{array}{c}{\varvec{v}}_{6}\\ {\varvec{v}}_{7}\\ \begin{array}{c}{\varvec{v}}_{8}\\ {\varvec{v}}_{9}\\ \begin{array}{c}{\varvec{v}}_{10}\\ {\varvec{v}}_{11}\\ \begin{array}{c}{\varvec{v}}_{12}\\ {\varvec{v}}_{13}\\ \begin{array}{c}⋮\\ ⋮\\ \begin{array}{c}⋮\\ ⋮\\ \begin{array}{c}{\varvec{v}}_{26}\\ {\varvec{v}}_{27}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\left[\begin{array}{cccccccccccccccccc}0& 1& 2& 3& 3& 4& 3& 2& 4& 5& 6& 5& 6& \dots & \dots & \dots & 9& 7\\ 1& 0& 1& 2& 2& 3& 2& 1& 3& 4& 5& 4& 5& \dots & \dots & \dots & 8& 6\\ 2& 1& 0& 1& 1& 2& 3& 2& 4& 5& 4& 3& 4& \dots & \dots & \dots & 7& 5\\ 3& 2& 1& 0& 2& 3& 4& 3& 5& 6& 5& 4& 5& \cdots & \cdots & \cdots & 8& 6\\ 3& 2& 1& 2& 0& 1& 2& 3& 3& 4& 3& 2& 3& \cdots & \cdots & \cdots & 6& 4\\ 4& 3& 2& 3& 1& 0& 1& 2& 2& 3& 2& 1& 2& \cdots & \cdots & \cdots & 5& 3\\ 3& 2& 3& 4& 2& 1& 0& 1& 1& 2& 3& 2& 3& \cdots & \cdots & \cdots & 6& 4\\ 2& 1& 2& 3& 3& 2& 1& 0& 2& 3& 4& 3& 4& \cdots & \cdots & \cdots & 7& 5\\ 4& 3& 4& 5& 3& 2& 1& 2& 0& 1& 2& 3& 4& \cdots & \cdots & \cdots & 5& 3\\ 5& 4& 5& 6& 4& 3& 2& 3& 1& 0& 1& 2& 3& \cdots & \cdots & \cdots & 4& 2\\ 6& 5& 4& 5& 3& 2& 3& 4& 2& 1& 0& 1& 2& \cdots & \cdots & \cdots & 3& 1\\ 5& 4& 3& 4& 2& 1& 2& 3& 3& 2& 1& 0& 1& \ddots & \ddots & \ddots & ⋮& ⋮\\ 6& 5& 4& 5& 3& 2& 3& 4& 4& 3& 2& 1& 0& \ddots & \ddots & \ddots & ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & \ddots & \ddots & ⋮& ⋮\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & ⋮& ⋮\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & ⋮& ⋮\\ 9& 8& 7& 8& 6& 5& 6& 7& 5& 4& 3& 4& 5& \cdots & \cdots & \cdots & 0& 2\\ 7& 6& 5& 6& 4& 3& 4& 5& 3& 2& 1& 2& 3& \cdots & \cdots & \cdots & 2& 0\end{array}\right]\) The Wiener index of Vitamin B2 is calculated as (by using (1)) . \(W\left({G}_{2}\right)=\frac{3396}{2}=1698\) (5) The Hyper Wiener index of vitamin B2 is calculated as (by using (2)) $$WW\left({G}_{2}\right)=\frac{1}{2}\left(20564+3396\right)=11980$$ 6 2.3 Wiener and Hyper Wiener Indices for Vitamin B3 Niacin, or vitamin B3, is a water-soluble member of the B-complex group of vitamins. It is essential for several metabolic functions, including the synthesis of energy and the preservation of the health of the neurological, digestive, and skin systems. The chemical formula for Vitamin B3 is C6H5NO2. Figure 5 depicts the molecular structure of Vitamin B3, while Fig. 6 represents its molecular graph, denoted as G3. Graph G3 comprises 9 vertices and 9 edges. The distance matrix of Vitamin B3 is $$\begin{array}{ccc}{ \varvec{v}}_{1}& { \varvec{v}}_{2}& \begin{array}{ccc}{ \varvec{v}}_{3}& {\varvec{v}}_{4 }& \begin{array}{ccc}{\varvec{v}}_{5 }& {\varvec{v}}_{6}& \begin{array}{ccc}{\varvec{v}}_{7 }& {\varvec{v}}_{8}& { \varvec{v}}_{9}\end{array}\end{array}\end{array}\end{array}$$ $$\begin{array}{c}\begin{array}{c}\begin{array}{c}{\varvec{v}}_{1}\\ {\varvec{v}}_{2}\end{array}\\ {\varvec{v}}_{3}\end{array}\\ {\varvec{v}}_{4}\\ \begin{array}{c}{\varvec{v}}_{5}\\ {\varvec{v}}_{6}\\ \begin{array}{c}{\varvec{v}}_{7}\\ {\varvec{v}}_{8}\\ {\varvec{v}}_{9}\end{array}\end{array}\end{array}\left[\begin{array}{ccccccccc}0& 1& 2& 3& 2& 1& 3& 2& 3\\ 1& 0& 1& 2& 3& 2& 4& 3& 4\\ 2& 1& 0& 1& 2& 3& 5& 4& 5\\ 3& 2& 1& 0& 1& 2& 4& 3& 4\\ 2& 3& 2& 1& 0& 1& 3& 2& 3\\ 1& 2& 3& 2& 1& 0& 2& 1& 2\\ 3& 4& 5& 4& 3& 2& 0& 1& 2\\ 2& 3& 4& 3& 2& 1& 1& 0& 1\\ 3& 4& 5& 4& 3& 2& 2& 1& 0\end{array}\right]$$ The Wiener index of Vitamin B3 is calculated by using Eq. ( 1 ) $$W\left({G}_{3}\right)=\frac{176}{2}=88$$ 7 The Hyper Wiener index of Vitamin B3 is calculated by using Eq. ( 2 ) $$WW\left({G}_{3}\right)=\frac{1}{2}\left(528+176\right)=352$$ 8 2.4 Wiener and Hyper Wiener Indices for Vitamin B6 Pyridoxine, another name for vitamin B6, is a water-soluble vitamin that is necessary for immunological response, brain development, and the metabolism of lipids, carbs, and proteins The chemical formula for Vitamin B6 is C8H11NO3. Figure 7 illustrates the chemical structure of Vitamin B6, while Fig. 8 represents its molecular graph, denoted as G4. Graph G4 consists of 12 vertices and 12 edges. The distance matrix of Vitamin B6 is $$\begin{array}{ccc}{ \varvec{v}}_{1}& { \varvec{v}}_{2}& \begin{array}{ccc}{ \varvec{v}}_{3}& {\varvec{v}}_{4 }& \begin{array}{ccc}{\varvec{v}}_{5}& { \varvec{v}}_{6}& \begin{array}{ccc}{ \varvec{v}}_{7 }& {\varvec{v}}_{8}& \begin{array}{cc}{ \varvec{v}}_{9 }& \begin{array}{cc}{\varvec{v}}_{10}& \begin{array}{cc}{ \varvec{v}}_{11 }& {\varvec{v}}_{12}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}$$ $$\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}{\varvec{v}}_{1}\\ {\varvec{v}}_{2}\end{array}\\ {\varvec{v}}_{3}\end{array}\\ {\varvec{v}}_{4}\end{array}\\ {\varvec{v}}_{5}\end{array}\\ {\varvec{v}}_{6}\end{array}\\ {\varvec{v}}_{7}\\ \begin{array}{c}{\varvec{v}}_{8}\\ {\varvec{v}}_{9}\\ \begin{array}{c}{\varvec{v}}_{10}\\ {\varvec{v}}_{11}\\ {\varvec{v}}_{12}\end{array}\end{array}\end{array}\left[\begin{array}{cccccccccccc}0& 1& 2& 3& 4& 4& 5& 5& 4& 3& 4& 5\\ 1& 0& 1& 2& 3& 3& 4& 4& 3& 2& 3& 4\\ 2& 1& 0& 1& 2& 2& 3& 3& 2& 1& 2& 3\\ 3& 2& 1& 0& 1& 1& 2& 2& 3& 2& 3& 4\\ 4& 3& 2& 1& 0& 2& 3& 3& 4& 3& 4& 5\\ 4& 3& 2& 1& 2& 0& 1& 1& 2& 3& 4& 5\\ 5& 4& 3& 2& 3& 1& 0& 2& 3& 4& 5& 6\\ 5& 4& 3& 2& 3& 1& 2& 0& 1& 2& 3& 4\\ 4& 3& 2& 3& 4& 2& 3& 1& 0& 1& 2& 3\\ 3& 2& 1& 2& 3& 3& 4& 2& 1& 0& 1& 2\\ 4& 3& 2& 3& 4& 4& 5& 3& 2& 1& 0& 1\\ 5& 4& 3& 4& 5& 5& 6& 4& 3& 2& 1& 0\end{array}\right]$$ The Wiener index of Vitamin B6 is calculated by using Eq. 1 is $$W\left({G}_{4}\right)=\frac{372}{2}=186$$ 9 The Hyper Wiener index of vitamin B6 is calculated by using Eq. 1 is $$WW\left({G}_{4}\right)=\frac{1}{2}\left(1264+372\right)=818$$ 10 Theorem 2. 5. Prove that the molecular Structure of Vitamins B1,B2,B3 and B6 satisfy the conditions of a metric spaced based on the wiener index and the associated distance matrix. Proof Let G i , where i = 1,2,3,4 the molecular graph of Vitamin B i for i = 1,2,3 and 6. Let D(G i ) denote the distance matrix associated with G i and let W i be the Wiener index of G i . The Structures of vitamins B1,B2,B3,B6 are represented by their molecular graph G 1 ,G 2 ,G 3 ,G 4 (Refer Fig. 2 , 4 , 6 , 8 ). The Wiener Index is defined by W(G)= \(\frac{1}{2}\sum _{i=1}^{n}\sum _{j=1}^{n}{d}_{ij}\) , where \({d}_{ij}\) is the distance between atoms V i and V j in the molecular graph G i .The distance \({d}_{ij}\) are obtained from the distance matrix D(G i ). The Distance matrix D(G i ) satisfies the conditions of a metric space Non negativity: \({d}_{ij}\ge 0\) for all i and j Identity of indiscernible: \({d}_{ij}\) =0 if and only if i = j. Symmetry: \({d}_{ij}\) = \({d}_{ji}\) for all i and j Triangle Inequality: \({d}_{ik}\le {d}_{ij}\) + \({d}_{jk}\) for all I, j and k. Therefore, based on the definition of the Wiener index and the properties of the distance matrix, the molecular structures of Vitamins B1, B2, B3, and B6 satisfies the conditions of a metric space. This completes the proof, establish the metric space properties of the molecular structure of the vitamins. Proposition 2.6 For the molecular graph of vitamin B i , Where i = 1,2,3 and 6 if the eccentricity index of a graph is n then the dominating number of the graph is n-1. Proof Let G i where i = 1,2,3,4 represents the molecular graph of Vitamin B i for i = 1,2,3 and 6.(Refer Fig. 2 , 4 , 6 , 8 ) Let G 4 be the molecular graph of Vitamin B 6 . (Fig. 8 ) V= { V 1 , V 2 ,V 3 ,V 4 ,V 5 ,V 6 ,V 7 ,V 8 ,V 9 ,V 10 ,V 11, V 12 } Let e(G i ) where i = 1,2,3,4 be the eccentricity index of the vitamins B i . The eccentricity of V is the greatest distance from V to any other vertex. The eccentricity index e(G 4 ) of Vitamin B6 is 6, from vertex v 12 to v 7 . Let \(\gamma\) (G i ) denote the dominating number of the vitamins B i . Case (i) Consider the Dominating Set D 1 = { \({ \text{V}}_{2}\) , \({ \text{V}}_{4},{ \text{V}}_{6},{ \text{V}}_{9},{ \text{V}}_{11}\) } V-D 1 ={ \({ \text{V}}_{1}\) , \({ \text{V}}_{3},{ \text{V}}_{5},{ \text{V}}_{7},{ \text{V}}_{8},{ \text{V}}_{10},{ \text{V}}_{12}\) } V 2 is connected to v 1 ,v 3 and is not linked to other vertices V 4 is connected to v 5 ,v 6 , v 3 and is not linked to any other vertices V 6 is connected to v 7 ,v 8 ,v 4 and is not linked to any other vertices V 9 is connected to v 8 ,v 10 and is not linked to any other vertices V 11 is connectedto v 10 ,v 12 and is not linked to any other vertices These vertices do not share any direct connections with each other. Case (ii) Consider the Dominating Set D 2 = { \({ \text{V}}_{1}\) , \({ \text{V}}_{4},{ \text{V}}_{6},{ \text{V}}_{10},{ \text{V}}_{12}\) } V-D 2 ={ \({ \text{V}}_{2}\) , \({ \text{V}}_{3},{ \text{V}}_{5},{ \text{V}}_{7},{ \text{V}}_{8},{ \text{V}}_{9},{ \text{V}}_{11}\) } V 1 is connected to v 2 and is not linked to any other vertices V 4 is connected to v 3 ,v 5 ,v 6 and is not linked to any other vertices V 6 is connected to v 7 ,v 8 ,v 4 and is not linked to any other vertices V 10 is connected to v 3 ,v 9 , v 11 and is not linked to any other vertices V 12 is connected to v 11 and is not linked to any other vertices These vertices do not share any direct connections with each other. From the two cases the dominating number \(\gamma\) (G 4 ) for vitamin B 6 is 5. Similarly , For the vitamin B 1, e(G 1 ) = 11 and \(\gamma\) (G 1 ) = 10 For the vitamin B 2, e(G 2 ) = 11 and \(\gamma\) (G 2 ) = 10. For the vitamin B 3, e(G 3 ) = 5 and \(\gamma\) (G 3 ) = 4. In General, if the eccentricity index of the graph is n then the dominating number of the graph is n-1. Results and Discussion In this section, the main computational results are presented. 3.1 Relationship between Wiener index, Hyper wiener index and Molecular weight of Vitamins Let W i represent the wiener index and WW i represent the Hyper Wiener index of Vitamins B i for i = 1,2,3 and 6. Let MW i represent the molecular weight of vitamin B i . Molecular weights are sourced from PubChem online. Table 1 displays information on the Wiener index, hyper Wiener index, and molecular weights of vitamins.Refer Eq. (3)-Eq. ( 10 ) Table 1 Wiener Index, Hyper Wiener Index, and Molecular Weights of Vitamins Vitamins W i WW i MW i Vitamin B1 642 4202 300.81 Vitamin B2 1698 11980 376.4 Vitamin B3 88 352 123.11 Vitamin B6 186 818 169.18 Figure 9 illustrates the graphical representation of the Wiener index for vitamins, Fig. 10 showcases the graph for the hyper Wiener index, and Fig. 11 exhibits the molecular weight. Furthermore, Fig. 12 visually demonstrates the interconnections among these parameters. In these graphs, the Red line corresponds to the Wiener index, the blue line corresponds to the hyper Wiener index, and the green line corresponds to the molecular weight. W i >WW i for all Vitamin B i . The smallest wiener index corresponds to the smallest molecular weight. W 3 < W 1 , W 3 < W 2, W 3 < W 6 and MW 3 < MW 1 , MW 3 < MW 2, MW 3 W 1 , W 2 > W 3 , W 2 > W 6 and MW 2 > MW 1 , MW 2 > MW 3 , MW 2 > MW 6 These expressions capture the relationships between Wiener indices, hyper Wiener indices, and molecular weights for the given set of vitamins.Similarly, for other B-complex vitamins like B5, B7, B9 and B12, we can calculate the indices and explore their relationships. Conclusion In this study, a focused analysis was conducted on four essential B-complex vitamins—B1, B2, B3, and B6—among the broader group of eight. Distance-based topological indices, specifically the Wiener and hyper Wiener indices, were systematically calculated. The graphical representations effectively illustrate the intricate relationships existing among these indices and the molecular weight of the selected vitamins. The established relationships and comparisons serve as a valuable resource in cheminformatics. They provide insights into potential applications and implications for drug development and related fields. This valuable resource offers a comprehensive understanding of the connections between molecular structures, topological indices, and physical properties. This study lays a foundation for further research exploring the connections between molecular structures, topological indices, and physical properties in the realm of B-complex vitamins and pharmaceutical compounds. Declarations Author Contribution T.G conducted the computation and analysed the results, while G.J supervised the findings . Author’s Declaration -Conflicts of Interest : None References NenadTrinajstic, Chemical Graph Theory . 2nd ed. Boca Raton; CRC Press,Inc,Florida.2000, Chap. 10, Topological Indices; pp. 240–242 I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry ,SpringerVerlag,Berlin, 1986 D. Bonchev, Information Theoretic Indices For Characterization of Chemical Structures , vol. 5 (Research Studies, New York, 1983) R. García-Domenech, J. Galvez, de J. Julian-Ortiz, Vicente, Pogliani, Lionello,Some New Trends in Chemical Graph Theory. Chem. Rev. 108 , 1127–1169 (2008) S.M. Suresh Elumalai, T. Hosamanib, Mansourc, Mohammad Ali Rostami,More on Inverse Degree and Topological Indices of Graphs. Filomat. 32 , 1 (2018) Sorgun, Sezer&Küçük, Hakan&Birgin, Kahraman, Some Distance-Based Topological Indices of Certain Polysaccharide. J. Mol. Struct. 1250 , 1–9 (2022) A. Mofidi, On Dominating Graph of Graphs, Median Graphs, Partial Cubes and Complement of Minimal Dominating Sets. Graphs Combinatorics 39 , 5(2023) L. Qiu, Li, Jianping&Jianbin, Zhang. On the eccentricity energy and eccentricity spectral radius of graphs with odd diameter. RAIRO - Oper. Res. 57 , 6(2023) Jayalalitha,Gopalakrishnan, Raji,&Senthil,S,Hyper Wiener Index of Molecular Graph of Naphthalene Using Domination. Int. J. Anal. experimental modalanalysis. 11 , 126–129 (2019) M. Hanna, E. Jaqua, V. Nguyen, Clay, J. B Vitamins: Funct. Uses Med. ThePermanenteJournal. 26 , 89–97 (2022) H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69 (1), 17–20 (1947) Brueckler, Franka&Doslic, Tomislav&Graovac, Ante &Gutman, Ivan,On a class of distance-based molecular structure descriptors. Chem. Phys. Lett. 503 , 336–338 (2011) K.C. Das, S. Elumalai, A. Ghosh, ToufikMansour,On conjecture of Merrifield–Simmons index. Discrete Appl Math. 288 , 211–217 (2021) Peng, Zhi-Ba &Nizami, Abdul & Iqbal, Zaffar&Munir, Mobeen& Muhammad, Hafiz & Ahmed,Waqar& Liu, Jia-Bao, “ Wiener and Hyper-Wiener Indices of Polygonal Cylinder and Torus” Complexity. 2021. S. Mondal, N. De, Pal, Anita, Topological Indices of Some Chemical Structures Applied for the Treatment of COVID-19 Patients. Polycycl. Aromat. Compd. 42 , 11(2020) Nagy, Benedek,The hyper-Wiener Index of diamond nanowires. Int. J. Quantum Chem. 124 , 1 (2023) Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4402009","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":303905928,"identity":"a30cec4a-6568-427f-a8c6-8034c6569a1a","order_by":0,"name":"T. Greeta","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAklEQVRIiWNgGAWjYFADCQbGBwkVQAYzcwNh1QeAmEeCgdngwxmQFkbitbBJzmwDcQlokY8+fPDxh5pt8vbS7Q+keefVRvO3A7X8qNiGU4vhubRkgwPHbhv2yJwxMObddjx3xmHGBsaeM7dxa+nhMZM4wHabsUcihyGZd9ux3AagFmbGNnxa+L//OPDvtn2PRPqDw7xzjuXOJ6RFnoeHjeFg2+3EHokEw8aZDTW5GwhpMeBhM5Y423c7uedGjjHDh2MHcjcCtRzE5xf5HuaHHyq+3bZtn5H+/EdCTV3uvPOHDz74UYHHlgOo/MNg8gCGOmRbGlD5dfgUj4JRMApGwQgFAHXAYctrIIDFAAAAAElFTkSuQmCC","orcid":"","institution":"VELS Institute of Science, Technology and Advanced Studies","correspondingAuthor":true,"prefix":"","firstName":"T.","middleName":"","lastName":"Greeta","suffix":""},{"id":303905929,"identity":"c431e28a-b163-4a86-ba09-90871bd0b1a7","order_by":1,"name":"G. Jayalalitha","email":"","orcid":"","institution":"VELS Institute of Science, Technology and Advanced Studies","correspondingAuthor":false,"prefix":"","firstName":"G.","middleName":"","lastName":"Jayalalitha","suffix":""}],"badges":[],"createdAt":"2024-05-10 17:15:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4402009/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4402009/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":56853182,"identity":"b97a59b8-cf75-4578-9817-0e3b97106db7","added_by":"auto","created_at":"2024-05-21 09:29:37","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":20089,"visible":true,"origin":"","legend":"\u003cp\u003eVitamin B1\u003c/p\u003e","description":"","filename":"Figure1.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/0a15ece0cde3a91c1a461ee4.png"},{"id":56853188,"identity":"67edc789-da9b-49a5-bd0c-7c21cbf0e809","added_by":"auto","created_at":"2024-05-21 09:29:37","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":19623,"visible":true,"origin":"","legend":"\u003cp\u003eMolecular Graph of Vitamin B1(Graph G1)\u003c/p\u003e","description":"","filename":"Figure2.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/6cc52b974429c91ceeb0a789.png"},{"id":56853908,"identity":"b246ccff-052e-481f-a779-2c93baf88c8f","added_by":"auto","created_at":"2024-05-21 09:37:37","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":27653,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eVitamin B2\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure3.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/44692587df91949fcbdbc047.png"},{"id":56853907,"identity":"01262ff1-a135-463d-a861-500697eb5ead","added_by":"auto","created_at":"2024-05-21 09:37:37","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":42582,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMolecular Graph of Vitamin B2 (Graph G2)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure4.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/6d2298af710355ebba9f8f1e.png"},{"id":56853185,"identity":"5a55b69f-0602-4120-a581-a707bed82ab1","added_by":"auto","created_at":"2024-05-21 09:29:37","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":7190,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eVitamin B3\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure5.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/d1b21f22be3f52861b2bcf83.png"},{"id":56853184,"identity":"ff455f3b-70cc-448d-a6ba-38d403b21166","added_by":"auto","created_at":"2024-05-21 09:29:37","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":11071,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMolecular Graph of Vitamin B3 (Graph G\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e3\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure6.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/96e66f1b38b768fdb4bf4d46.png"},{"id":56854448,"identity":"7a666e8a-e345-4a11-9ab4-b65083a455ca","added_by":"auto","created_at":"2024-05-21 09:45:37","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":14925,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eVitamin B6\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure7.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/8326a426aa8b3b25b6e92002.png"},{"id":56853910,"identity":"3e4fd652-bbfb-4b0a-93e5-ae8e9da8174a","added_by":"auto","created_at":"2024-05-21 09:37:38","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":13694,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMolecular Graph of Vitamin B4(Graph G\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e4\u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003e)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure8.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/d34b3b3439480e913e22e5a6.png"},{"id":56853912,"identity":"72e5afdd-9340-44f1-a620-0847fa1d270c","added_by":"auto","created_at":"2024-05-21 09:37:38","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":32857,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eWiener Index ofVitamins\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure9.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/9fab98674d584ea1c1ab601a.png"},{"id":56853189,"identity":"3d2a9300-aad0-43de-ab1b-02a75b3c6542","added_by":"auto","created_at":"2024-05-21 09:29:38","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":36618,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eHyper Wiener Index of Vitamins\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure10.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/7d39703c4ec5477340327096.png"},{"id":56854449,"identity":"0c10ed56-606c-43d3-80b8-b4c53b4919f2","added_by":"auto","created_at":"2024-05-21 09:45:38","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":31406,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMolecular Weights of Vitamins\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure11.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/f3876490a5eef27fad7c13fa.png"},{"id":56854987,"identity":"d761bc16-ad2b-4073-9299-21823dfe2fdc","added_by":"auto","created_at":"2024-05-21 09:53:38","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":63295,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eInterrelationship among Wiener index, Hyper wiener index and Molecular weight of vitamins\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Figure12.png","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/ff809b94d75d0ec6dc59cee7.png"},{"id":57145250,"identity":"5465a0e6-aadf-4ba2-849b-92a1d74086e2","added_by":"auto","created_at":"2024-05-25 18:46:38","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1589438,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4402009/v1/d73a45fb-2292-4554-aba8-de022b95fef1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Computation of Wiener and Hyper Wiener Indices in Several B-Complex Vitamins","fulltext":[{"header":"Introduction","content":"\u003cp\u003eIn a graph G=(V,E), the vertices in V represent the collection of points denoting atoms, and E signifies the cluster of connections (bonds) that illustrate the linking of atoms within the molecule[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. The significant contributions made in the domain of chemical graph theory have led to recent innovations and developments in chemical graph simulation and mathematical modelling. Chemical graph theory is crucial for understanding the structure of chemical compounds. Scientific studies consistently show that the properties of chemicals and medications are closely tied to their molecular makeup. In this theory, atoms are like points (vertices), and bonds are like connections (edges) between these points [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eTopological indices, numerical representations of the structural aspects of chemical substances, are crucial in chemical graph theory. These indices, often referred to as graph invariant indices, provide quantitative insights into the topology of chemical structures. They serve a vital contribution in extracting quantitative structure-property or structure-activity relationships (QSPR/QSAR) [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. These indices can be used independently or in conjunction with other quantitative variables to model and predict various properties or activities of chemical compounds. Chemical graph theory finds extensive implementations in pharmaceutical innovation, contributing significantly to the understanding and advancement of these fields.\u003c/p\u003e \u003cp\u003eDistance-based topological indices are mathematical measures obtained from the structural graph of a chemical substance [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. These indices capture important structural information by considering the distances between pairs of particles within the compounds. These indices play a crucial role in quantitative structure-activity relationship (QSAR) studies, cheminformatics, and computational chemistry.\u003c/p\u003e \u003cp\u003eIn the realm of graph theory, the dominating number focuses on vertex dominance. A dominating set in a graph comprises a subset of vertices, where each vertex either belongs to the dominating set or is adjacent to a vertex in the dominating set. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. γ(G), the dominating number, indicates the size, i.e., the number of vertices, in the smallest dominating set within a given graph. Alternatively, graph eccentricity evaluates the longest path between two vertices, also known as the greatest distance or furthest separation between any pair of vertices in the graph. [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eRaji and Jayalalitha [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] conducted a study in which they explored the hyper Wiener index of the molecular graph of naphthalene, employing a domination-based approach.\u003c/p\u003e \u003cp\u003eThe Vitamin B complex, consisting of a group of water-soluble vitamins, is essential for numerous physiological functions such as energy metabolism, nerve function, and cell division. B vitamins, when combined with other micronutrients, are vital for maintaining overall health and well-being. These vitamins significantly influence energy levels, cognitive function, and cellular metabolism [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Comprising eight distinct vitamins, this complex is renowned for its diverse and vital contributions to human health. In this paper, our attention is directed toward the initial quartet of B vitamins\u0026mdash;B1, B2, B3, and B6.\u003c/p\u003e \u003cp\u003eThe aim of this study is to compute topological indices for the four B-vitamins. The investigation includes the exploration of various theorems derived from these indices. Ultimately, a graphical representation is presented to illustrate the correlation between the computed indices and the physical properties of the vitamins.\u003c/p\u003e \u003cp\u003eIn Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the materials and methods used for the analysis are detailed. Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e2.1\u003c/span\u003e focuses on calculating the Wiener and hyper-Wiener indices for vitamin B1 using the Distance Matrix method. Section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e describes the calculation of these indices for vitamin B2, also using the Distance Matrix method. Section \u003cspan refid=\"Sec5\" class=\"InternalRef\"\u003e2.3\u003c/span\u003e deals with the calculation for vitamin B3, again utilizing the Distance Matrix. Section \u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003e2.4\u003c/span\u003e explains the method for calculating the Wiener and hyper-Wiener indices of vitamin B6 using the Distance Matrix. The proof for Theorem 2.5 and Proposition \u003cspan refid=\"FPar3\" class=\"InternalRef\"\u003e2.6\u003c/span\u003e is presented. Moving to Section \u003cspan refid=\"Sec7\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the results of the analysis are discussed. Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e3.1\u003c/span\u003e specifically explores the correlations between the physical properties of the vitamins and their respective topological indices.\u003c/p\u003e"},{"header":"Materials and Methods","content":"\u003cp\u003eSimple graphs are used as models for the representation of B-vitamins. Certainly, the four B-complex vitamins under consideration for analysis are Vitamin B1 (Thiamine), Vitamin B2 (Riboflavin), Vitamin B3 (Niacin), and Vitamin B6 (Pyridoxine).\u003c/p\u003e\n\u003cp\u003eMain findings revolve around employing distance-based topological indices for several B-vitamins. The chemical structures of Vitamin B1, B2, B3, and B6 were obtained from online sources. In our investigation, we specifically concentrated on molecular graphs of these compounds where hydrogen atoms were omitted[\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e]. This choice was deliberate as vertices representing hydrogen atoms were excluded due to their negligible contribution to graph isomorphism.\u003c/p\u003e\n\u003cp\u003eThe Wiener index stands as the inaugural generation of topological indices, a concept pioneered by the chemist Harry Wiener in 1947[\u003cspan class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e12\u003c/span\u003e]. In his seminal work, Wiener introduced the term \u003cem\u003epath number\u003c/em\u003e for graphs, representing the total of distances between pairs of carbon atoms within molecules. This innovative concept marked the inception of quantitative measures to capture the molecular structure's inherent connectivity patterns, laying the foundation for subsequent advancements in the field of chemical graph theory. In their paper titled \"\u003cem\u003eOn Conjecture of Merrified Simmons Index\u003c/em\u003e ,\" Suresh Elumalai and colleagues provided a clear explanation of this matter[\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e]. The Wiener Index is defined as half of the total of distances between every pair of atoms in a molecular graph [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e]. Mathematically, for a graph G with n vertices (atoms)\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$$W\\left(G\\right)=\\frac{1}{2}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{d}_{ij}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}\\)\u003c/span\u003e\u003c/span\u003edenotes the minimum path length between vertices i and j within the molecular graph. The Wiener index is often used in cheminformatics and quantitative structure-activity relationship (QSAR) studies [\u003cspan class=\"CitationRef\"\u003e15\u003c/span\u003e] to predict molecular properties based on molecular structure.\u003c/p\u003e\n\u003cp\u003eThe Hyper Wiener Index is an extension of the Wiener index and was introduced to incorporate additional structural information [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e]. It is defined as\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ2\" class=\"mathdisplay\"\u003e$$WW\\left(G\\right)=\\frac{1}{2}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{\\left[d\\right({v}_{i},{v}_{j})}^{2}+d\\left({v}_{i},{v}_{j}\\right)]$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe Hyper Wiener index provides a more detailed measure of connectivity patterns within a molecule by considering both the distance from atom i to atom j and the distance from atom j to atom i. It enhances the representation of molecular structure and has applications in quantitative structure-activity relationship studies.\u003c/p\u003e\n\u003cp\u003eWhen transforming the molecular structure into a molecular graph, atoms are translated into vertices, and the bonds between atoms are represented as edges [\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e]. Let G\u003csub\u003e1\u003c/sub\u003e denote the molecular graph of Vitamin B1, G\u003csub\u003e2\u003c/sub\u003e represent the molecular graph of Vitamin B2, G\u003csub\u003e3\u003c/sub\u003e symbolize the molecular graph of Vitamin B3, and G\u003csub\u003e4\u003c/sub\u003e stand for the molecular graph of Vitamin B6.\u003c/p\u003e\n\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n\u003ch2\u003e2.1 Wiener and Hyper Wiener Indices for Vitamin B1\u003c/h2\u003e\n\u003cp\u003eThiamine, or vitamin B1, is a water-soluble vitamin that is a member of the B-complex family. As a cofactor for enzymes involved in the transformation of carbohydrates into energy, it is essential to energy metabolism.\u003c/p\u003e\n\u003cp\u003eThe chemical formula for Vitamin B1 is C12H17N4OS. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e represents the chemical structure of Vitamin B1, and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e depicts its molecular graph, denoted as Graph G1. Graph G1 has 18 vertices and 19 edges.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe distance matrix for vitamin B1 is\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\begin{array}{ccc}{ \\varvec{v}}_{1 }\u0026amp; {\\varvec{v}}_{2} \u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{3}\u0026amp; {\\varvec{v}}_{4}\u0026amp; \\begin{array}{ccc} {\\varvec{v}}_{5}\u0026amp; { \\varvec{v}}_{6}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{7 }\u0026amp; {\\varvec{v}}_{8}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{9 }\u0026amp; {\\varvec{v}}_{10 } \u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{11}\u0026amp; { \\varvec{v}}_{12 }\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{13 }\u0026amp; {\\varvec{v}}_{14 }\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{15 }\u0026amp; {\\varvec{v}}_{16}\u0026amp; \\begin{array}{cc}{ \\varvec{v}}_{17}\u0026amp; {\\varvec{v}}_{18}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\begin{array}{c}{\\varvec{v}}_{1}\\\\ {\\varvec{v}}_{2}\\\\ \\begin{array}{c}{\\varvec{v}}_{3}\\\\ {\\varvec{v}}_{4}\\\\ \\begin{array}{c}{\\varvec{v}}_{5}\\\\ {\\varvec{v}}_{6}\\\\ \\begin{array}{c}{\\varvec{v}}_{7}\\\\ {\\varvec{v}}_{8}\\\\ \\begin{array}{c}{\\varvec{v}}_{9}\\\\ {\\varvec{v}}_{10}\\\\ \\begin{array}{c}{\\varvec{v}}_{11}\\\\ {\\varvec{v}}_{12}\\\\ \\begin{array}{c}{\\varvec{v}}_{13}\\\\ {\\varvec{v}}_{14}\\\\ \\begin{array}{c}{\\varvec{v}}_{15}\\\\ {\\varvec{v}}_{16}\\\\ \\begin{array}{c}{\\varvec{v}}_{17}\\\\ {\\varvec{v}}_{18}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\left[\\begin{array}{cccccccccccccccccc}0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 6\u0026amp; 6\u0026amp; 5\u0026amp; 7\u0026amp; 8\u0026amp; 9\\\\ 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 5\u0026amp; 5\u0026amp; 4\u0026amp; 6\u0026amp; 7\u0026amp; 8\\\\ 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 6\u0026amp; 6\u0026amp; 5\u0026amp; 7\u0026amp; 8\u0026amp; 9\\\\ 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 7\u0026amp; 7\u0026amp; 6\u0026amp; 8\u0026amp; 9\u0026amp; 10\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 8\u0026amp; 8\u0026amp; 8\u0026amp; 7\u0026amp; 9\u0026amp; 10\u0026amp; 11\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 6\u0026amp; 6\u0026amp; 5\u0026amp; 7\u0026amp; 8\u0026amp; 9\\\\ 3\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 5\u0026amp; 5\u0026amp; 4\u0026amp; 6\u0026amp; 7\u0026amp; 8\\\\ 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 4\u0026amp; 4\u0026amp; 3\u0026amp; 5\u0026amp; 6\u0026amp; 7\\\\ 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 3\u0026amp; 2\u0026amp; 4\u0026amp; 5\u0026amp; 6\\\\ 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 2\u0026amp; 1\u0026amp; 3\u0026amp; 4\u0026amp; 5\\\\ 5\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 4\\\\ 6\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 8\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 3\u0026amp; 4\u0026amp; 5\\\\ 6\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 8\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\\\\ 6\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 8\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\\\\ 5\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 3\u0026amp; 4\u0026amp; 5\\\\ 7\u0026amp; 6\u0026amp; 7\u0026amp; 8\u0026amp; 9\u0026amp; 7\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 0\u0026amp; 1\u0026amp; 2\\\\ 8\u0026amp; 7\u0026amp; 8\u0026amp; 9\u0026amp; 10\u0026amp; 8\u0026amp; 7\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 1\u0026amp; 0\u0026amp; 1\\\\ 9\u0026amp; 8\u0026amp; 9\u0026amp; 10\u0026amp; 11\u0026amp; 9\u0026amp; 8\u0026amp; 7\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 2\u0026amp; 1\u0026amp; 0\\end{array}\\right]\\)\u003c/span\u003e \u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eThe Wiener index of Vitamin B\u003csub\u003e1\u003c/sub\u003e is calculated as (by using Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e))\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(W\\left({G}_{1}\\right)=\\frac{1286}{2}\\)\u003c/span\u003e \u003c/span\u003e = 643 (3)\u003c/p\u003e\n\u003cp\u003eThe Hyper Wiener index of vitamin B\u003csub\u003e1\u003c/sub\u003e is calculated as (by using Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e))\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ3\" class=\"mathdisplay\"\u003e$$WW\\left({G}_{1}\\right)=\\frac{1}{2}\\left(7120+1286\\right)=4202$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n\u003ch2\u003e2.2 Wiener and Hyper Wiener Indices for Vitamin B2\u003c/h2\u003e\n\u003cp\u003eVitamin B2, or Riboflavin, is an essential water-soluble vitamin that is a part of the B-complex family. It helps turn food into energy by being an essential component of several metabolic processes. Riboflavin is essential for the maintenance of healthy skin, eyes, and nerve functions.\u003c/p\u003e\n\u003cp\u003eThe chemical formula for Vitamin B2 is C\u003csub\u003e17\u003c/sub\u003eH\u003csub\u003e20\u003c/sub\u003eN\u003csub\u003e4\u003c/sub\u003eO\u003csub\u003e6\u003c/sub\u003e. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e represents the molecular structure of Vitamin B2. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e illustrates the molecular graph of Vitamin B2, denoted as G2. Graph G2 consists of 27 vertices and 29 edges.\u003c/p\u003e\n\u003cp\u003eThe distance matrix of Vitamin B2 is\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\begin{array}{ccc}{ \\varvec{v}}_{1 } \u0026amp; {\\varvec{v}}_{2}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{3}\u0026amp; {\\varvec{v}}_{4}\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{5} \u0026amp; {\\varvec{v}}_{6}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{7}\u0026amp; {\\varvec{v}}_{8} \u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{9}\u0026amp; { \\varvec{v}}_{10 }\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{11 }\u0026amp; {\\varvec{v}}_{12}\u0026amp; \\begin{array}{cc}{\\varvec{v}}_{13 }\u0026amp; \\begin{array}{ccc}\\cdots \u0026amp; \\cdots \u0026amp; \\begin{array}{cc}{ \\varvec{v}}_{26 }\u0026amp; {\\varvec{v}}_{27}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\begin{array}{c}\\begin{array}{c}{ \\varvec{v}}_{1 }\\\\ {\\varvec{v}}_{2}\\end{array}\\\\ {\\varvec{v}}_{3}\\\\ \\begin{array}{c}{\\varvec{v}}_{4}\\\\ {\\varvec{v}}_{5}\\\\ \\begin{array}{c}{\\varvec{v}}_{6}\\\\ {\\varvec{v}}_{7}\\\\ \\begin{array}{c}{\\varvec{v}}_{8}\\\\ {\\varvec{v}}_{9}\\\\ \\begin{array}{c}{\\varvec{v}}_{10}\\\\ {\\varvec{v}}_{11}\\\\ \\begin{array}{c}{\\varvec{v}}_{12}\\\\ {\\varvec{v}}_{13}\\\\ \\begin{array}{c}⋮\\\\ ⋮\\\\ \\begin{array}{c}⋮\\\\ ⋮\\\\ \\begin{array}{c}{\\varvec{v}}_{26}\\\\ {\\varvec{v}}_{27}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\left[\\begin{array}{cccccccccccccccccc}0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 5\u0026amp; 6\u0026amp; \\dots \u0026amp; \\dots \u0026amp; \\dots \u0026amp; 9\u0026amp; 7\\\\ 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 4\u0026amp; 5\u0026amp; \\dots \u0026amp; \\dots \u0026amp; \\dots \u0026amp; 8\u0026amp; 6\\\\ 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 4\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; \\dots \u0026amp; \\dots \u0026amp; \\dots \u0026amp; 7\u0026amp; 5\\\\ 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 3\u0026amp; 5\u0026amp; 6\u0026amp; 5\u0026amp; 4\u0026amp; 5\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 8\u0026amp; 6\\\\ 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 6\u0026amp; 4\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 5\u0026amp; 3\\\\ 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 6\u0026amp; 4\\\\ 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 7\u0026amp; 5\\\\ 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 5\u0026amp; 3\\\\ 5\u0026amp; 4\u0026amp; 5\u0026amp; 6\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 4\u0026amp; 2\\\\ 6\u0026amp; 5\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 3\u0026amp; 1\\\\ 5\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; ⋮\u0026amp; ⋮\\\\ 6\u0026amp; 5\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; ⋮\u0026amp; ⋮\\\\ ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; ⋮\u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; ⋮\u0026amp; ⋮\\\\ \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; ⋮\u0026amp; ⋮\\\\ \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; \\ddots \u0026amp; ⋮\u0026amp; ⋮\\\\ 9\u0026amp; 8\u0026amp; 7\u0026amp; 8\u0026amp; 6\u0026amp; 5\u0026amp; 6\u0026amp; 7\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 0\u0026amp; 2\\\\ 7\u0026amp; 6\u0026amp; 5\u0026amp; 6\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; \\cdots \u0026amp; 2\u0026amp; 0\\end{array}\\right]\\)\u003c/span\u003e \u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eThe Wiener index of Vitamin B2 is calculated as (by using (1))\u003c/p\u003e\n\u003cp\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(W\\left({G}_{2}\\right)=\\frac{3396}{2}=1698\\)\u003c/span\u003e\u003c/span\u003e (5)\u003c/p\u003e\n\u003cp\u003eThe Hyper Wiener index of vitamin B2 is calculated as (by using (2))\u003c/p\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ4\" class=\"mathdisplay\"\u003e$$WW\\left({G}_{2}\\right)=\\frac{1}{2}\\left(20564+3396\\right)=11980$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n\u003ch2\u003e2.3 Wiener and Hyper Wiener Indices for Vitamin B3\u003c/h2\u003e\n\u003cp\u003eNiacin, or vitamin B3, is a water-soluble member of the B-complex group of vitamins. It is essential for several metabolic functions, including the synthesis of energy and the preservation of the health of the neurological, digestive, and skin systems.\u003c/p\u003e\n\u003cp\u003eThe chemical formula for Vitamin B3 is C6H5NO2. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e depicts the molecular structure of Vitamin B3, while Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e represents its molecular graph, denoted as G3. Graph G3 comprises 9 vertices and 9 edges.\u003c/p\u003e\n\u003cp\u003eThe distance matrix of Vitamin B3 is\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equa\" class=\"mathdisplay\"\u003e$$\\begin{array}{ccc}{ \\varvec{v}}_{1}\u0026amp; { \\varvec{v}}_{2}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{3}\u0026amp; {\\varvec{v}}_{4 }\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{5 }\u0026amp; {\\varvec{v}}_{6}\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{7 }\u0026amp; {\\varvec{v}}_{8}\u0026amp; { \\varvec{v}}_{9}\\end{array}\\end{array}\\end{array}\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equb\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}\\begin{array}{c}\\begin{array}{c}{\\varvec{v}}_{1}\\\\ {\\varvec{v}}_{2}\\end{array}\\\\ {\\varvec{v}}_{3}\\end{array}\\\\ {\\varvec{v}}_{4}\\\\ \\begin{array}{c}{\\varvec{v}}_{5}\\\\ {\\varvec{v}}_{6}\\\\ \\begin{array}{c}{\\varvec{v}}_{7}\\\\ {\\varvec{v}}_{8}\\\\ {\\varvec{v}}_{9}\\end{array}\\end{array}\\end{array}\\left[\\begin{array}{ccccccccc}0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 3\u0026amp; 2\u0026amp; 3\\\\ 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 4\u0026amp; 3\u0026amp; 4\\\\ 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 5\u0026amp; 4\u0026amp; 5\\\\ 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 4\u0026amp; 3\u0026amp; 4\\\\ 2\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 3\u0026amp; 2\u0026amp; 3\\\\ 1\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 1\u0026amp; 2\\\\ 3\u0026amp; 4\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 2\\\\ 2\u0026amp; 3\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 1\u0026amp; 0\u0026amp; 1\\\\ 3\u0026amp; 4\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 2\u0026amp; 1\u0026amp; 0\\end{array}\\right]$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe Wiener index of Vitamin B3 is calculated by using Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e)\u003c/p\u003e\n\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ5\" class=\"mathdisplay\"\u003e$$W\\left({G}_{3}\\right)=\\frac{176}{2}=88$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe Hyper Wiener index of Vitamin B3 is calculated by using Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e)\u003c/p\u003e\n\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ6\" class=\"mathdisplay\"\u003e$$WW\\left({G}_{3}\\right)=\\frac{1}{2}\\left(528+176\\right)=352$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n\u003ch2\u003e2.4 \u003cem\u003eWiener and Hyper Wiener Indices for Vitamin B6\u003c/em\u003e\u003c/h2\u003e\n\u003cp\u003ePyridoxine, another name for vitamin B6, is a water-soluble vitamin that is necessary for immunological response, brain development, and the metabolism of lipids, carbs, and proteins\u003c/p\u003e\n\u003cp\u003eThe chemical formula for Vitamin B6 is C8H11NO3. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e illustrates the chemical structure of Vitamin B6, while Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e represents its molecular graph, denoted as G4. Graph G4 consists of 12 vertices and 12 edges.\u003c/p\u003e\n\u003cp\u003eThe distance matrix of Vitamin B6 is\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$$\\begin{array}{ccc}{ \\varvec{v}}_{1}\u0026amp; { \\varvec{v}}_{2}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{3}\u0026amp; {\\varvec{v}}_{4 }\u0026amp; \\begin{array}{ccc}{\\varvec{v}}_{5}\u0026amp; { \\varvec{v}}_{6}\u0026amp; \\begin{array}{ccc}{ \\varvec{v}}_{7 }\u0026amp; {\\varvec{v}}_{8}\u0026amp; \\begin{array}{cc}{ \\varvec{v}}_{9 }\u0026amp; \\begin{array}{cc}{\\varvec{v}}_{10}\u0026amp; \\begin{array}{cc}{ \\varvec{v}}_{11 }\u0026amp; {\\varvec{v}}_{12}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}\\end{array}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equd\" class=\"mathdisplay\"\u003e$$\\begin{array}{c}\\begin{array}{c}\\begin{array}{c}\\begin{array}{c}\\begin{array}{c}\\begin{array}{c}{\\varvec{v}}_{1}\\\\ {\\varvec{v}}_{2}\\end{array}\\\\ {\\varvec{v}}_{3}\\end{array}\\\\ {\\varvec{v}}_{4}\\end{array}\\\\ {\\varvec{v}}_{5}\\end{array}\\\\ {\\varvec{v}}_{6}\\end{array}\\\\ {\\varvec{v}}_{7}\\\\ \\begin{array}{c}{\\varvec{v}}_{8}\\\\ {\\varvec{v}}_{9}\\\\ \\begin{array}{c}{\\varvec{v}}_{10}\\\\ {\\varvec{v}}_{11}\\\\ {\\varvec{v}}_{12}\\end{array}\\end{array}\\end{array}\\left[\\begin{array}{cccccccccccc}0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 4\u0026amp; 5\u0026amp; 5\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\\\\ 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 4\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\\\\ 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\\\\ 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 1\u0026amp; 2\u0026amp; 2\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\\\\ 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 0\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 6\\\\ 5\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 2\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 4\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 3\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\u0026amp; 3\\\\ 3\u0026amp; 2\u0026amp; 1\u0026amp; 2\u0026amp; 3\u0026amp; 3\u0026amp; 4\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\u0026amp; 2\\\\ 4\u0026amp; 3\u0026amp; 2\u0026amp; 3\u0026amp; 4\u0026amp; 4\u0026amp; 5\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\u0026amp; 1\\\\ 5\u0026amp; 4\u0026amp; 3\u0026amp; 4\u0026amp; 5\u0026amp; 5\u0026amp; 6\u0026amp; 4\u0026amp; 3\u0026amp; 2\u0026amp; 1\u0026amp; 0\\end{array}\\right]$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe Wiener index of Vitamin B6 is calculated by using Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e is\u003c/p\u003e\n\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ7\" class=\"mathdisplay\"\u003e$$W\\left({G}_{4}\\right)=\\frac{372}{2}=186$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe Hyper Wiener index of vitamin B6 is calculated by using Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e is\u003c/p\u003e\n\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ8\" class=\"mathdisplay\"\u003e$$WW\\left({G}_{4}\\right)=\\frac{1}{2}\\left(1264+372\\right)=818$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eTheorem 2.\u003c/strong\u003e\u003cstrong\u003e5.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eProve that the molecular Structure of Vitamins B1,B2,B3 and B6 satisfy the conditions of a metric spaced based on the wiener index and the associated distance matrix.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProof\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eLet G\u003csub\u003ei\u003c/sub\u003e, where i\u0026thinsp;=\u0026thinsp;1,2,3,4 the molecular graph of Vitamin B\u003csub\u003ei\u003c/sub\u003e for i\u0026thinsp;=\u0026thinsp;1,2,3 and 6.\u003c/p\u003e\n\u003cp\u003eLet D(G\u003csub\u003ei\u003c/sub\u003e ) denote the distance matrix associated with G\u003csub\u003ei\u003c/sub\u003e and let W\u003csub\u003ei\u003c/sub\u003e be the Wiener index of G\u003csub\u003ei\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003eThe Structures of vitamins B1,B2,B3,B6 are represented by their molecular graph G\u003csub\u003e1\u003c/sub\u003e,G\u003csub\u003e2\u003c/sub\u003e,G\u003csub\u003e3\u003c/sub\u003e,G\u003csub\u003e4\u003c/sub\u003e (Refer Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e,\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e,\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e,\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003eThe Wiener Index is defined by W(G)= \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{1}{2}\\sum _{i=1}^{n}\\sum _{j=1}^{n}{d}_{ij}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}\\)\u003c/span\u003e\u003c/span\u003e is the distance between atoms V\u003csub\u003ei\u003c/sub\u003e and V\u003csub\u003ej\u003c/sub\u003e in the molecular graph G\u003csub\u003ei\u003c/sub\u003e.The distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}\\)\u003c/span\u003e\u003c/span\u003eare obtained from the distance matrix D(G\u003csub\u003ei\u003c/sub\u003e).\u003c/p\u003e\n\u003cp\u003eThe Distance matrix D(G\u003csub\u003ei\u003c/sub\u003e ) satisfies the conditions of a metric space\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eNon negativity: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}\\ge 0\\)\u003c/span\u003e\u003c/span\u003e for all i and j\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIdentity of indiscernible:\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}\\)\u003c/span\u003e\u003c/span\u003e=0 if and only if i\u0026thinsp;=\u0026thinsp;j.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSymmetry:\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}\\)\u003c/span\u003e\u003c/span\u003e=\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ji}\\)\u003c/span\u003e\u003c/span\u003e for all i and j\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eTriangle Inequality:\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ik}\\le {d}_{ij}\\)\u003c/span\u003e\u003c/span\u003e +\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{jk}\\)\u003c/span\u003e\u003c/span\u003e for all I, j and k.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eTherefore, based on the definition of the Wiener index and the properties of the distance matrix, the molecular structures of Vitamins B1, B2, B3, and B6 satisfies the conditions of a metric space.\u003c/p\u003e\n\u003cp\u003eThis completes the proof, establish the metric space properties of the molecular structure of the vitamins.\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eProposition 2.6\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFor the molecular graph of vitamin B\u003csub\u003ei\u003c/sub\u003e, Where i\u0026thinsp;=\u0026thinsp;1,2,3 and 6 if the eccentricity index of a graph is n then the dominating number of the graph is n-1.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eProof\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eLet G\u003csub\u003ei\u003c/sub\u003e where i\u0026thinsp;=\u0026thinsp;1,2,3,4 represents the molecular graph of Vitamin B\u003csub\u003ei\u003c/sub\u003e for i\u0026thinsp;=\u0026thinsp;1,2,3 and 6.(Refer Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e,\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e,\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e,\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e)\u003c/p\u003e\n\u003cp\u003eLet G\u003csub\u003e4\u003c/sub\u003e be the molecular graph of Vitamin B\u003csub\u003e6\u003c/sub\u003e. (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e)\u003c/p\u003e\n\u003cp\u003eV= { V\u003csub\u003e1\u003c/sub\u003e, V\u003csub\u003e2\u003c/sub\u003e ,V\u003csub\u003e3\u003c/sub\u003e ,V\u003csub\u003e4\u003c/sub\u003e ,V\u003csub\u003e5\u003c/sub\u003e ,V\u003csub\u003e6\u003c/sub\u003e ,V\u003csub\u003e7\u003c/sub\u003e ,V\u003csub\u003e8\u003c/sub\u003e ,V\u003csub\u003e9\u003c/sub\u003e ,V\u003csub\u003e10\u003c/sub\u003e ,V\u003csub\u003e11,\u003c/sub\u003e V\u003csub\u003e12\u003c/sub\u003e }\u003c/p\u003e\n\u003cp\u003eLet e(G\u003csub\u003ei\u003c/sub\u003e) where i\u0026thinsp;=\u0026thinsp;1,2,3,4 be the eccentricity index of the vitamins B\u003csub\u003ei\u003c/sub\u003e. The eccentricity of V is the greatest distance from V to any other vertex.\u003c/p\u003e\n\u003cp\u003eThe eccentricity index e(G\u003csub\u003e4\u003c/sub\u003e ) of Vitamin B6 is 6, from vertex v\u003csub\u003e12\u003c/sub\u003e to v\u003csub\u003e7\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003eLet \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e(G\u003csub\u003ei\u003c/sub\u003e ) denote the dominating number of the vitamins B\u003csub\u003ei\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCase (i)\u003c/strong\u003e Consider the Dominating Set D\u003csub\u003e1\u003c/sub\u003e= {\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{2}\\)\u003c/span\u003e\u003c/span\u003e ,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{4},{ \\text{V}}_{6},{ \\text{V}}_{9},{ \\text{V}}_{11}\\)\u003c/span\u003e\u003c/span\u003e }\u003c/p\u003e\n\u003cp\u003eV-D\u003csub\u003e1\u003c/sub\u003e={\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{1}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{3},{ \\text{V}}_{5},{ \\text{V}}_{7},{ \\text{V}}_{8},{ \\text{V}}_{10},{ \\text{V}}_{12}\\)\u003c/span\u003e\u003c/span\u003e}\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e2\u003c/sub\u003e is connected to v\u003csub\u003e1\u003c/sub\u003e ,v\u003csub\u003e3\u003c/sub\u003eand is not linked to other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e4\u003c/sub\u003eis connected to v\u003csub\u003e5\u003c/sub\u003e ,v\u003csub\u003e6\u003c/sub\u003e, v\u003csub\u003e3\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e6\u003c/sub\u003e is connected to v\u003csub\u003e7\u003c/sub\u003e ,v\u003csub\u003e8\u003c/sub\u003e,v\u003csub\u003e4\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e9\u003c/sub\u003e is connected to v\u003csub\u003e8\u003c/sub\u003e,v\u003csub\u003e10\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e11\u003c/sub\u003e is connectedto v\u003csub\u003e10\u003c/sub\u003e ,v\u003csub\u003e12\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eThese vertices do not share any direct connections with each other.\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003eCase (ii)\u003c/strong\u003e Consider the Dominating Set D\u003csub\u003e2\u003c/sub\u003e= {\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{1}\\)\u003c/span\u003e\u003c/span\u003e ,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{4},{ \\text{V}}_{6},{ \\text{V}}_{10},{ \\text{V}}_{12}\\)\u003c/span\u003e\u003c/span\u003e}\u003c/p\u003e\n\u003cp\u003eV-D\u003csub\u003e2\u003c/sub\u003e={\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{2}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\text{V}}_{3},{ \\text{V}}_{5},{ \\text{V}}_{7},{ \\text{V}}_{8},{ \\text{V}}_{9},{ \\text{V}}_{11}\\)\u003c/span\u003e\u003c/span\u003e }\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e1\u003c/sub\u003e is connected to v\u003csub\u003e2\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e4\u003c/sub\u003e is connected to v\u003csub\u003e3\u003c/sub\u003e ,v\u003csub\u003e5\u003c/sub\u003e,v\u003csub\u003e6\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e6\u003c/sub\u003e is connected to v\u003csub\u003e7\u003c/sub\u003e,v\u003csub\u003e8\u003c/sub\u003e,v\u003csub\u003e4\u003c/sub\u003eand is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e10\u003c/sub\u003e is connected to v\u003csub\u003e3\u003c/sub\u003e ,v\u003csub\u003e9\u003c/sub\u003e, v\u003csub\u003e11\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eV\u003csub\u003e12\u003c/sub\u003e is connected to v\u003csub\u003e11\u003c/sub\u003e and is not linked to any other vertices\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eThese vertices do not share any direct connections with each other.\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eFrom the two cases the dominating number \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e(G\u003csub\u003e4\u003c/sub\u003e) for vitamin B\u003csub\u003e6\u003c/sub\u003e is 5.\u003c/p\u003e\n\u003cp\u003eSimilarly ,\u003c/p\u003e\n\u003cp\u003eFor the vitamin B\u003csub\u003e1,\u003c/sub\u003e e(G\u003csub\u003e1\u003c/sub\u003e )\u0026thinsp;=\u0026thinsp;11 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e(G\u003csub\u003e1\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;10\u003c/p\u003e\n\u003cp\u003eFor the vitamin B\u003csub\u003e2,\u003c/sub\u003e e(G\u003csub\u003e2\u003c/sub\u003e )\u0026thinsp;=\u0026thinsp;11 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e(G\u003csub\u003e2\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;10.\u003c/p\u003e\n\u003cp\u003eFor the vitamin B\u003csub\u003e3,\u003c/sub\u003e e(G\u003csub\u003e3\u003c/sub\u003e )\u0026thinsp;=\u0026thinsp;5 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e(G\u003csub\u003e3\u003c/sub\u003e)\u0026thinsp;=\u0026thinsp;4.\u003c/p\u003e\n\u003cp\u003eIn General, if the eccentricity index of the graph is n then the dominating number of the graph is n-1.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"Results and Discussion","content":"\u003cp\u003eIn this section, the main computational results are presented.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Relationship between Wiener index, Hyper wiener index and Molecular weight of Vitamins\u003c/h2\u003e \u003cp\u003eLet W\u003csub\u003ei\u003c/sub\u003e represent the wiener index and WW\u003csub\u003ei\u003c/sub\u003e represent the Hyper Wiener index of Vitamins B\u003csub\u003ei\u003c/sub\u003e for i\u0026thinsp;=\u0026thinsp;1,2,3 and 6.\u003c/p\u003e \u003cp\u003eLet MW\u003csub\u003ei\u003c/sub\u003e represent the molecular weight of vitamin B\u003csub\u003ei\u003c/sub\u003e. Molecular weights are sourced from PubChem online.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e displays information on the Wiener index, hyper Wiener index, and molecular weights of vitamins.Refer Eq.\u0026nbsp;(3)-Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e10\u003c/span\u003e)\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eWiener Index, Hyper Wiener Index, and Molecular Weights of Vitamins\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVitamins\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eW\u003csub\u003ei\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eWW\u003csub\u003ei\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMW\u003csub\u003ei\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eVitamin B1\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e642\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4202\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e300.81\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eVitamin B2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1698\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e11980\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e376.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eVitamin B3\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e88\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e352\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e123.11\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eVitamin B6\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e186\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e818\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e169.18\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e illustrates the graphical representation of the Wiener index for vitamins, Fig.\u0026nbsp;10 showcases the graph for the hyper Wiener index, and Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e11\u003c/span\u003e exhibits the molecular weight. Furthermore, Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e12\u003c/span\u003e visually demonstrates the interconnections among these parameters. In these graphs, the Red line corresponds to the Wiener index, the blue line corresponds to the hyper Wiener index, and the green line corresponds to the molecular weight.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eW\u003csub\u003ei\u003c/sub\u003e\u0026gt;WW\u003csub\u003ei\u003c/sub\u003e for all Vitamin B\u003csub\u003ei\u003c/sub\u003e .\u003c/p\u003e \u003cp\u003eThe smallest wiener index corresponds to the smallest molecular weight.\u003c/p\u003e \u003cp\u003eW\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;W\u003csub\u003e1\u003c/sub\u003e, W\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;W\u003csub\u003e2,\u003c/sub\u003e W\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;W\u003csub\u003e6\u003c/sub\u003e and MW\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;MW\u003csub\u003e1\u003c/sub\u003e, MW\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;MW\u003csub\u003e2,\u003c/sub\u003e MW\u003csub\u003e3\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;MW\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e \u003cp\u003eThe Largest wiener index corresponds to the largest molecular weight.\u003c/p\u003e \u003cp\u003eW\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;W\u003csub\u003e1\u003c/sub\u003e, W\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;W\u003csub\u003e3\u003c/sub\u003e, W\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;W\u003csub\u003e6\u003c/sub\u003e and MW\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;MW\u003csub\u003e1\u003c/sub\u003e, MW\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;MW\u003csub\u003e3\u003c/sub\u003e, MW\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;\u0026gt;\u0026thinsp;MW\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e \u003cp\u003eThese expressions capture the relationships between Wiener indices, hyper Wiener indices, and molecular weights for the given set of vitamins.Similarly, for other B-complex vitamins like B5, B7, B9 and B12, we can calculate the indices and explore their relationships.\u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn this study, a focused analysis was conducted on four essential B-complex vitamins\u0026mdash;B1, B2, B3, and B6\u0026mdash;among the broader group of eight. Distance-based topological indices, specifically the Wiener and hyper Wiener indices, were systematically calculated. The graphical representations effectively illustrate the intricate relationships existing among these indices and the molecular weight of the selected vitamins. The established relationships and comparisons serve as a valuable resource in cheminformatics. They provide insights into potential applications and implications for drug development and related fields. This valuable resource offers a comprehensive understanding of the connections between molecular structures, topological indices, and physical properties. This study lays a foundation for further research exploring the connections between molecular structures, topological indices, and physical properties in the realm of B-complex vitamins and pharmaceutical compounds.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthor Contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eT.G conducted the computation and analysed the results, while G.J supervised the findings\u003cstrong\u003e.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor\u0026rsquo;s Declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e-Conflicts of Interest : None\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eNenadTrinajstic, \u003cem\u003eChemical Graph Theory\u003c/em\u003e. 2nd ed. Boca Raton; CRC Press,Inc,Florida.2000, Chap. 10, Topological Indices; pp. 240\u0026ndash;242\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eI. Gutman, O.E. Polansky, \u003cem\u003eMathematical Concepts in Organic Chemistry\u003c/em\u003e,SpringerVerlag,Berlin, 1986\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eD. Bonchev, \u003cem\u003eInformation Theoretic Indices For Characterization of Chemical Structures\u003c/em\u003e, vol. 5 (Research Studies, New York, 1983)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eR. Garc\u0026iacute;a-Domenech, J. Galvez, de J. Julian-Ortiz, Vicente, Pogliani, Lionello,Some New Trends in Chemical Graph Theory. Chem. Rev. \u003cb\u003e108\u003c/b\u003e, 1127\u0026ndash;1169 (2008)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eS.M. Suresh Elumalai, T. Hosamanib, Mansourc, Mohammad Ali Rostami,More on Inverse Degree and Topological Indices of Graphs. Filomat. \u003cb\u003e32\u003c/b\u003e, 1 (2018)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSorgun, Sezer\u0026amp;K\u0026uuml;\u0026ccedil;\u0026uuml;k, Hakan\u0026amp;Birgin, Kahraman, Some Distance-Based Topological Indices of Certain Polysaccharide. J. Mol. Struct. \u003cb\u003e1250\u003c/b\u003e, 1\u0026ndash;9 (2022)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eA. Mofidi, On Dominating Graph of Graphs, Median Graphs, Partial Cubes and Complement of Minimal Dominating Sets. Graphs Combinatorics \u003cb\u003e39\u003c/b\u003e, 5(2023)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eL. Qiu, Li, Jianping\u0026amp;Jianbin, Zhang. On the eccentricity energy and eccentricity spectral radius of graphs with odd diameter. RAIRO - Oper. 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Compd. \u003cb\u003e42\u003c/b\u003e, 11(2020)\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNagy, Benedek,The hyper-Wiener Index of diamond nanowires. Int. J. Quantum Chem. \u003cb\u003e124\u003c/b\u003e, 1 (2023)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Chemical graph theory, Vitamin B Complex, Distance matrix, Dominating number, Topological indices","lastPublishedDoi":"10.21203/rs.3.rs-4402009/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4402009/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eB vitamins are crucial for the proper functioning of our body. Vitamin B complex, comprising eight essential vitamins, supports growth and sustains energy levels. Graph theory aids chemists in systematically molecular modeling structures and reactions. Topological indices serve as mathematical descriptors, analyzing physicochemical properties. In this work, four B complex Vitamins\u0026mdash;B1, B2, B3, and B6\u0026mdash;are examined. The investigation centers on utilizing the distance matrix. It computes both the Wiener Index and Hyper Wiener Index, providing valuable perceptions toward the structural attributes of these vitamins. The outcomes of these metrics are systematically compared with the respective molecular weights of the vitamins. This comparison offers a comprehensive understanding of their chemical characteristics. It delves into the relationship among the structural features, topological indices, and molecular weights of essential B-complex vitamins. This paper explores enhancing our understanding of their biochemical roles and potential applications in health and medicine.\u003c/p\u003e","manuscriptTitle":"Computation of Wiener and Hyper Wiener Indices in Several B-Complex Vitamins","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-21 09:29:33","doi":"10.21203/rs.3.rs-4402009/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"ea5ce6e8-c7e6-43cb-858d-c7bb1ef1cb45","owner":[],"postedDate":"May 21st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-05-25T18:38:30+00:00","versionOfRecord":[],"versionCreatedAt":"2024-05-21 09:29:33","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4402009","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4402009","identity":"rs-4402009","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}