Polynomials and Chemical Structures

The use of M-polynomials to represent chemical compounds and chemical networks is a revolutionary idea that results in topological Invariants These findings are utilized to link the chemical characteristics and bioactivities of chemical compounds and chemical networks.The M-polynomial, denoted as $M(X;x,y)$, is obtained by summing the products of the number of edges, denoted as $m_{pq}(X)$, with a specific property involving the coordinates of the vertices. More precisely, we consider edges $ab$ in the structure $X$ where the coordinates of the vertices $\zeta^a$ and $\zeta^b$ satisfies $\zeta^a,\zeta^b=(p,q)$, where $p$ and $q$ are both greater than or equal to 1. The M-polynomial is given by $M(X;x,y)=\sum\limits_{p\leq q}m_{pq}(X)x^{p}y^{q}$.\\ We will use a general approach based on topological polynomials to compute important properties of chemical structures, specifically the Trans-Pd-$(NH_2)S$ lattice $TPL[m,n]$ and Metal-Organic Super lattice $(MOS[m,n])$. These properties include various topological invariants such as the Zagreb indices, Randic indices, harmonic index, symmetric division index, forgotten index, and inverse index. We use M-polynomial, which counts the number of edges with specific vertex coordinates, to calculate these invariants.