A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Zn

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The ideal-based non-zero divisor graph ∅ I ( Z n ) , constructed from the ring of integers modulo n with respect to a proper ideal I . This graph extends the classical zero-divisor graph framework and serves as a visual and structural invariant for analyzing ideal interactions in finite commutative rings. Methods Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of ∅ I ( Z n ) . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function ϕ ( n ) . The analysis distinguishes cases based on the parity and primality of n , as well as the generator of I . Topological indices, including the Zagreb and Randić indices, are formulated to quantify structural complexity. Results We establish necessary and sufficient conditions for the connectivity of ∅ I ( Z n ) , proving it is connected for all n ≥ 10 and any non-zero proper ideal I . For prime n ∉ { 2 , 3 } , the graph is shown to be complete. General formulas are provided for calculating vertex degrees based on gcd ( x , d ) where I = . Furthermore, the structure and computation cut-sets are characterized for Z p 2 and composite n = xy . Moreover, the domination number γ ( ∅ I ( Z n ) )=1 and girth gr ( ∅ I ( Z n ) )=3 is established for n ≥ 10 . General expressions for Zagreb and Randić indices are derived, directly linking graph invariants to n and d . Conclusions The graph ∅ I ( Z n ) serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in Z n . These results establish systematic connections between ring-theoretic properties and graph parameters, enabling both qualitative and quantitative analysis through connectivity, degree distributions, cut-sets, and topological indices. 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F1000Research 2026, 15 :44 ( https://doi.org/10.12688/f1000research.172788.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] Sameer Kadem 1 , Ali Abd Aubad https://orcid.org/0000-0002-4764-2946 2 Sameer Kadem 1 , Ali Abd Aubad https://orcid.org/0000-0002-4764-2946 2 PUBLISHED 09 Jan 2026 Author details Author details 1 Department of Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, Iraq 2 Department of Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, Iraq Sameer Kadem Roles: Conceptualization, Data Curation, Formal Analysis, Funding Acquisition, Methodology, Resources, Writing – Original Draft Preparation, Writing – Review & Editing Ali Abd Aubad Roles: Conceptualization, Investigation, Project Administration, Supervision OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract Background The study of algebraic structures through graph-theoretic representations provides a powerful visual and combinatorial framework for analyzing ring-theoretic properties. The ideal-based non-zero divisor graph ∅ I ( Z n ) , constructed from the ring of integers modulo n with respect to a proper ideal I . This graph extends the classical zero-divisor graph framework and serves as a visual and structural invariant for analyzing ideal interactions in finite commutative rings. Methods Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of ∅ I ( Z n ) . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function ϕ ( n ) . The analysis distinguishes cases based on the parity and primality of n , as well as the generator of I . Topological indices, including the Zagreb and Randić indices, are formulated to quantify structural complexity. Results We establish necessary and sufficient conditions for the connectivity of ∅ I ( Z n ) , proving it is connected for all n ≥ 10 and any non-zero proper ideal I . For prime n ∉ { 2 , 3 } , the graph is shown to be complete. General formulas are provided for calculating vertex degrees based on gcd ( x , d ) where I = . Furthermore, the structure and computation cut-sets are characterized for Z p 2 and composite n = xy . Moreover, the domination number γ ( ∅ I ( Z n ) )=1 and girth gr ( ∅ I ( Z n ) )=3 is established for n ≥ 10 . General expressions for Zagreb and Randić indices are derived, directly linking graph invariants to n and d . Conclusions The graph ∅ I ( Z n ) serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in Z n . These results establish systematic connections between ring-theoretic properties and graph parameters, enabling both qualitative and quantitative analysis through connectivity, degree distributions, cut-sets, and topological indices. READ ALL READ LESS Keywords non-zero divisor graph, cut-set, connected, degree, topological graph indices, dominating number Corresponding Author(s) Ali Abd Aubad ( [email protected] ) Close Corresponding author: Ali Abd Aubad Competing interests: No competing interests were disclosed. Grant information: This research was partially supported by the 4th Annual International Conference on Information and Sciences (AICIS’25), organized by the University of Fallujah. Further details about the conference can be found at: https://edas.info/index.php?c=34198 No additional grants or financial support were received from public, commercial, or not-for-profit funding agencies for the research, authorship, or publication of this article. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Copyright: © 2026 Kadem S and Abd Aubad A. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Kadem S and Abd Aubad A. A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.12688/f1000research.172788.1 ) First published: 09 Jan 2026, 15 :44 ( https://doi.org/10.12688/f1000research.172788.1 ) Latest published: 09 Jan 2026, 15 :44 ( https://doi.org/10.12688/f1000research.172788.1 ) 1. Introduction By transforming algebraic structures into graphs, the algebraic properties can be visualized, hidden symmetries can be uncovered, and intricate relationships can be simplified. By revealing patterns and invariants, graph theory tools facilitate the intuitive study of zero-divisors, ideals, and ring structures. This helps with classification, problem solving, and computational exploration by bridging the gap between abstract algebra and combinatorics. For an example, see Refs. 1 – 7 . For the commutative ring R and a proper ideal I of R , Redmond 8 was the first to identify the ideal-based zero divisor graph, designated as Γ I ( R ) , which contains a vertex set { α ∈ I C : αβ ∈ I , for some β ∈ I C } , and two distinct vertices μ and λ are adjacent if μλ ∈ I . Inspired by the preceding definition, we construct a simple graph associated with a ring R and a proper ideal I of R , denoted as ∅ I ( R ) , With the exception of the ideal I , the multiplicative identity, and its additive inverse, every vertex represents an element of R. Moreover, an edge connects two different vertices only when their product (in either order) is not in the ideal I . This type of graph is known as a non-zero divisor graph when I = 0; various studies have investigated this instance. 9 – 11 In this study, we assume that all graphs are undirected and that, basic graphs are devoid of many edges or loops. A graph Γ is said to be connected if a path exists between every pair of vertices. The girth of the graph represents the length of the smallest cycle. When a graph does not have any cycles, its girth is considered as infinite. Subset D ⊆ V ( Γ ) is said to be the dominating set of a graph Γ if every vertex α ∈ D c is adjacent to at least one vertex in D. The maximal cardinality of the smallest set that dominates G is represented by the domination number, which is abbreviated as γ ( G ) . Additional definitions pertaining to graph theory can be found in Ref. 12 . In this study, we look at certain essential characteristics of the ∅ I ( Zn ) graphs, illustrating their connectivity, and formulate general equations for the cut sets, vertex degrees, and specific topological degree-based indices. The results delineate cut-Sets, establish generic criteria for connectedness, and formulate expressions for vertex degrees based on divisibility relations in ∅ I ( Zn ) . Moreover, many topological indices have been computed to assess symmetry and structural complexity. 2. Preliminaries This section first outlines the definition of the ideal-based non-zero divisor graph, followed by an illustration example. The subsequent outcomes of our investigation presented in the graph. Definition 2.1: Let R be a ring and I be a proper ideal for R . The ideal-based non-zero divisor graph, denoted by ∅ I ( R ), has vertex set V ( ∅ I ( R )) = R \ { I , 1 , − 1 }. Moreover, two different vertices δ and γ in V ( ∅ I ( R )) are adjacent if and only if either δγ ∉ I or γδ ∉ I. Example 2.2: Let R = Z 6 and let I 0 = { 0 } , I 1 = { 0 , 2 , 4 } , I 2 = { 0 , 3 } . The following Figure 1 are used to illustrate the graph. Lemma 2.3: If there is an invertible element α in V ( ∅ I ( R ) ) , then for any proper ideal I of a ring R , the graph ∅ I ( R ) is connected. Proof: Because α is invertible then there is β ∈ R , such that αβ = βα = 1 . Let γ ∈ V ( ( ∅ I ( R ) ) = R \ { I , 1 , − 1 } , if αγ ∈ I and γα ∈ I , then we have βαγ ∈ I and γαβ ∈ I , which is a contradiction. Thus ∅ I ( R ) is connected. Definition 2.4: 13 ϕ ( n ) denotes the number of positive integers that are co-prime to n , where n ≥ 1 . Theorem 2.5: 13 If the integer n > 1 has prime factorization n = P 1 S 1 P 2 S 2 … P r S r , then ϕ ( n ) = ∏ i = 1 r ( P i S i − P i S i − 1 ) . ϕ ( n ) is the number of invertible elements in the integer ring of module n , which is noteworthy. Lemma 2.6: For n > 6, ϕ ( n ) > 3. Proof: We shall partition the proof into two distinct cases. - Case 1: n is a prime number then ϕ ( n ) = n-1 > 3. - Case 2: if n = P 1 S 1 P 2 S 2 … P r S r (composite), where P 1 , P 2 , …, P r represent r different prime numbers. Thus, ϕ ( n ) = ∏ i = 1 r ( P i S i − P i S i − 1 ) = 3 if and only if j exists such that ( P j S j − P j S j − 1 ) = 3 and ∏ i = 1 i ≠ j r ( P i S i − P i S i − 1 ) = 1, this is impossible when n > 6. Theorem 2.7: 9 The non-zero divisor graph ∅ 0 ( Z n ) is connected if and only if n ∉ {1, 2, 3, 6}. Figure 1. Example for ∅ I ( R ). 3. Main results Connectivity of the graph ∅ I ( Z n ) is examined in this section for two different scenarios: n ≤ 9 and n ≥ 10. With the help of examples, we further explore the idea of a cut-set and how it is calculated for graph ∅ I ( Zn ). For n ∈ {1,2,3,6,8}, the Table 1 below provides information about the ideal-based non-zero divisor graph of Z n . Lemma 3.1: For any prime number n ∉ { 2 , 3 } , ∅ I ( Z n ) represents a completely connected graph. Proof: Given that n is a prime number, the only proper ideal for Z n is the zero ideal. According to Theorem 2.7 , ∅ I ( Z n ) is connected. Assume that γ , β ∈ V ( ∅ I ( Z n ) ) with γβ = 0 ; therefore, either γ = 0 or β = 0 , leading to a contradiction. This indicated that the graph was complete. Lemma 3.2: For any prime number p ≥ 2 , the graph ∅ I ( Z P 2 ) is connected. Proof: The proper ideal of the ring Z P 2 are I = { 0 } and J = ⟨ p ⟩ which is maximal ideal of Z P 2 . • Case 1: ∅ I ( Z P 2 ) is connected by Theorem 2.7 • Case 2: Assume that ∅ J ( Z P 2 ) disconnected graph, there exist γ , δ in different connected component meaning: There is no path between γ and δ equivalently, for all α ∈ V ( ∅ J ( Z P 2 )) = Z P 2 \ { J , 1 , − 1 } either αγ ∈ J or αδ ∈ J , consider the ideal K generated by J ∪ ⟨ γ ⟩ and since γ ∉ J , then J ⊊ K , contradiction. For the integer module n , Z n = { 1 , 2 , … , n − 1 } and the ideal I = d Z n = { 0 , d , 2 d , … , ( ( n / d ) − 1 ) d } for some integer d \ n . Since V ( ∅ I ( Zn ) ) = Z n \ { I , 1 , − 1 } , we can now develop a method to calculate the order of the graph ∅ I ( Z n ) as following: | V ( ∅ I ( Z n ) ) | = n − n d − 2 , where | I | = n d . Lemma 3.3: Let n be an odd, non-prime number then, the non-zero ideal of Z n with the greatest order is the ideal generated by the prime number p \ n , where p = r 2 + n − r , for some integer r such that r 2 + n ∈ Z . Proof: If n = 2 k + 1 is odd number (not prime), then the ideal has the largest of order is the ideal generated by prime number that divides n, we will provide several cases and present a general formula to determine the largest prime number p that divides n. we know that | V ( ∅ I ( Zn ) ) | = n − n p − 2 , where | I | = n p . Therefor if p \ ( n = 2 k + 1 ) , then k = p − 1 2 , ( p − 1 2 ) p , ( p − 1 2 ) p + p , ( p − 1 2 ) p + 2 p , ( p − 1 2 ) p + 3 p , … . Then, the general formula for p is p = r 2 + n − r for some integer r that makes r 2 + n ∈ Z . Example 3.4: If n = 15 take r = 1 then p = 3 is the greatest order of ideal I such that | V ( ∅ I ( Zn ) ) | = 15 − 15 3 − 2 = 8 . Theorem 3.5: For any integer n ≥ 10 , the graph ∅ I ( Z n ) is connected for every non-zero proper ideal I of Z n . Proof: The proof will be divided into two cases. Case 1: I is a prime ideal; let γ , δ ∈ V ( ∅ I ( Z n )), with γδ ∈ I , then either γ ∈ I or δ ∈ I , contradiction. Thus ∅ I ( Z n ) connected (complete). Case 2: I not prime ideal. Therefore, we have the following subcases. • If n is even number, then the ideal has the largest of order is the ideal generated by 2, | V ( ∅ I ( Z n ) ) | = n − n 2 − 2 = n 2 − 2 , that means when n ≥ 10 , then | V ( ∅ I ( Zn ) ) | ≥ 3. By Lemma 2.6 , then ϕ ( n ) > 3, and our result follows immediately by Lemma 2.3 . • If n = 2 k + 1 is an odd number (not prime), then by Lemma 3.3 , the ideal with the largest order is the ideal generated by prime number p that divides n, where p = r 2 + n − r , for some integer r such that r 2 + n ∈ Z . If n > 10 and n is an odd number, then the greatest order of ideal I guarantees | V ( ∅ I ( Z n ) ) | ≥ 3 , By Lemma 2.6 , then ϕ ( n ) > 3, and our result follows immediately by Lemma 2.3 . Corollary 3.5: For any integer n ≥ 10, then γ ( ∅ I ( Z n ) ) = 1 and gr ( ∅ I ( Z n ) ) = 3 . Proof: For any integer n ≥ 10 , from the previous proof of the theorem 3.3 then | V ( ∅ I ( Z n ) ) | ≥ 3 for any ideal I , and by Lemma 2.6 we have γ ( ∅ I ( Z n )) = 1 and gr ( ∅ I ( Z n ) ) = 3 . Table 1. Information about ∅ I ( Z n ), n ∈ {1,2,3,6,8}. n Ring Z n Ideal I Vertex set V ( ∅ I ( Z n )) Connectivity properties 1 Z 1 { 0 } Empty set 2 Z 2 { 0 } Empty set 3 Z 3 { 0 } Empty set 6 Z 6 I 0 = { 0 } I 1 = { 0 , 2 , 4 } I 2 = { 0 , 3 } V ( ∅ I 0 ( Z 6 )) = { 2 , 3 , 4 } , V ( ∅ I 1 ( Z 6 )) = { 2 , 4 } , V ( ∅ I 2 ( Z 6 )) = { 3 } ∅ I 0 ( Z 6 ) disconnected ∅ I 1 ( Z 6 ) connected ∅ I 2 ( Z 6 ) connected 8 Z 8 I 0 = { 0 } I 1 = { 0 , 2 , 4 , 6 } I 2 = { 0 , 4 } V ( ∅ I 0 ( Z 8 )) = { 2 , 3 , 4 , 5 , 6 , } V ( ∅ I 1 ( Z 8 )) = { 3 , 5 } V ( ∅ I 2 ( Z 8 )) = { 2 , 3 , 5 , 6 } ∅ I ( Z 8 ) connected for any ideal I 4. Cut-set of the graph ∅ I ( Z n ) Definition 4.1: 14 A cut-set is defined as a set of vertices { β , γ , δ , … } in a connected graph G , where G can be represented as the union of two subgraphs X and Y , such that 1. E ( X ) ≠ ∅ and E ( Y ) ≠ ∅ (the edges set of X and Y are not empty) 2. E ( X ) ∪ E ( Y ) = E ( G ) and V ( X ) ∪ V ( Y ) = V ( G ) 3. V ( X ) ∩ V ( Y ) = { β , γ , δ , … } 4. X \ { β , γ , δ , … } ≠ ∅ and Y \ { β , γ , δ , … } ≠ ∅ . Moreover, no proper subset of { β , γ , δ , … } also acts as a cut set for any choice of X and Y . Here, an example that illustrates the definition of the cut-set in graph Example 4.1: In the graph ∅ 0 ( Z 8 ) of ring Z 8 , the set C = { 3 , 5 } represents the cut set. As shown Figure 2 . V ( ∅ 0 ( Z 8 ) ) = { 2 , 3 , 4 , 5 , 6 } . Proposition 4.2: For every prime number p greater than 2, the cut-set of the non-zero divisor graph ∅ 0 ( Z P 2 ) is { α ∈ V ( ∅ 0 ( Z P 2 ) ) \ ann ( p ) } . Proof: ann ( P ) = { x ∈ Z P 2 | Px = 0 } = { p , 2 p , … , ( p − 1 ) p } , for x and y in ann ( p ) , then x = kp and y = hp , for some k , h ∈ Z P and xy = kh P 2 = 0 , it follows that x is not connected with y . The graph that excludes the vertices of set ann ( p ) is unconnected. This is minimal because any proper subset of C. C preserves the pathways between zero-divisors via units by leaving some units intact. Example 4.4: The cut set in the graph ∅ 0 ( Z 3 2 ) of the ring Z 3 2 is the set C = {2,4,5,7}, as illustrated in Figure 3 . Theorem 4.5: Let = x . y , where x ≠ y and x > y . Then, the Cut-set of Z n , for n > 4 , corresponds the set C = P n ∪ D n \ ( ⟨ x ⟩ ⋃ ⟨ y ⟩ ) , where P n = { p ∈ v ( ( Zn ) ) | p is prime residues of n } D n = { d ∈ v ( ( Zn ) ) | d is zero divisor of Zn } Proof: Consider the subgraphs X and Y , which are generated by the ideal ⟨ x ⟩ and ⟨ y ⟩ , respectively. Consequently, subgraphs X and Y have no edges between them if γ ∈ X and δ ∈ Y then, δ = 0 , Consequently, subgraphs X and Y have no edges between them. Now, if C ~ = C \ { a }, then either a ∈ P n , which means that a is invertible, and by Lemma 2.3 , a connect all vertices of ∅ 0 ( Z n ) contradiction, or a ∈ D n , then there exists path from x to y by a , since if ax ≡ 0 ( mod n ) then ax = kn , k ∈ Z , thus kny = axy , therefore a = xy ∈ ⟨ y ⟩ , a contradiction. Example 4.3: For the ring Z 12 , n = 3.4 , then the cut set of Zn , for n > 4 , corresponds to the set C = P n ∪ D n \ ( ⟨ x ⟩ ⋃ ⟨ y ⟩ ) = {3,7,210}, as illustrated in the Figure 4 , where P n = { 5 , 7 } and D n = { 2 , 10 } . Figure 2. Cut-set of ∅ 0 ( Z 8 ). Figure 3. The cut set of ∅ 0 ( Z 3 2 ) . Figure 4. The Cut-set of ∅ 0 ( Z 12 ) . 5. The degree of a vertex x in ∅ I ( Z n ) Definition 5.1: Let x be a vertex of any graph; the degree of x is represented as deg ( x ) , and the degree of x is defined as follows: deg ( x ) = | { y ∈ V \ { x } : d ( x , y ) = 1 } | , ( d ( x , y ) length of the short path). From the above definition, it is clear that deg ( x ) = deg ( − x ) i n ∅ I ( Z n ) Theorem 5.2: Let R = Z n be the ring of integers module, and let ∅ I ( Z n ) denoted the ideal based non-zero divisor graph of R . Suppose I = ⟨ d ⟩ is ideal of, where d \ n . for any vertex x ∈ V ( ∅ I ( R )) and g = gcd ( x , d ) . Then, the degree of x is given by deg ( x ) = { n − n d − 3 if g = 1 n − n g − 2 if g > 1 Proof: Case 1: if g = gcdgcd ( x , d ) = 1 , assume that for all y ∈ V ( ∅ I ( R )), xy ∈ I = ⟨ d ⟩ , thus d \ xy , but gcd ( x , d ) = 1 , therefore, d \ y , that means y ∈ I , a contradiction. Then, by the definition of an ideal-based non-zero divisor, y must be a universal vertex, so deg ( x ) = | V ( ∅ I ( R ) ) | − 1 = n − 2 − | I | − 1 = n − 3 − n d . Case 2: if g = gcd ( x , d ) > 1 , let x = gl and d = gτ , so we need to exclude the ideal generated by τ , when I want an account for deg ( x ). But deg ( x ) = n − 2 − | ⟨ τ ⟩ | = n − 2 − n g . Example 5.3: Applying Theorem 5.2 , we will present a Table 2 , illustrating examples of calculating the degree of a vertex in certain ∅ I ( Z n ) . Table 2. Calculating the vertex degree in some ∅ I ( Z n ). n I = ⟨ d ⟩ d x gcd ( x , d ) = g deg ( x ) 12 I = ⟨ 4 ⟩ 4 5 1 n − n d − 3 = 7 12 I = ⟨ 4 ⟩ 4 10 2 n − n g − 2 = 4 18 I = ⟨ 9 ⟩ 9 15 3 n − n g − 2 = 10 18 I = ⟨ 9 ⟩ 9 10 1 n − n d − 3 = 13 30 I = ⟨ 2 ⟩ 2 7 1 n − n d − 3 = 12 30 I = ⟨ 6 ⟩ 6 21 3 n − n g − 2 = 18 Corollary 5.4: Let = Z P 2 , P > 2 , prime number, for any vertex x ∈ V ( ∅ ⟨ P ⟩ ( R )). The degree of x is then given by d e g ( x ) = P 2 − P − 3 . Corollary 5.5: Let = Z 2 P , P > 2 , prime number, for any vertex x ∈ V ( ∅ I ( R )), For any non-zero ideal of R , then 1. deg ( x ) = P − 3 if x ∈ V ( ∅ ⟨ 2 ⟩ ( R )) 2. deg ( x ) = 2 P − 5 if x ∈ V ( ∅ ⟨ P ⟩ ( R )) Theorem 5.6: Let R = Z n be the ring of integers module n , and let ∅ 0 ( Z n ) be a non-zero divisor graph of R . Then, for any vertex x ∈ V ( ∅ 0 ( R ) ) , set g = deg ( x , n ). Thus, the degree of x is given by: deg ( x ) = { n − 4 if g = 1 n − g − 3 if x 2 ≠ 0 n − g − 2 if x 2 = 0 Proof: To prove the result, we should consider the following subcases : Case 1: if g = gcd ( x , n ) = 1 , assume that for all y ∈ V ( ∅ 0 ( R ) ) , xy = 0 , but x is invertible thus, y = 0 , which is a contradiction. Therefore deg ( x ) = | V ( ∅ 0 ( R ) ) | − 1 = n − 3 − 1 . Case 2: Let S = { y ∈ V ( ∅ 0 ( R ) ) : xy = 0 } , representing the non-neighbors of x , then S = ann ( x ) ∩ V ( ∅ 0 ( R ) ) ,which contains exactly g elements if x 2 ≠ 0 , and g + 1 elements when x 2 = 0 Example 5.7: Let R = Z 12 then the following Table 3 , representing the degree of every vertex x of the graph ∅ 0 ( R ) . Example 5.8: Let R = Z 30 then the following Table 4 , representing the degree of every vertex x of the graph ∅ 0 ( R ) . Table 3. The vertex degrees of ∅ 0 ( Z 12 ). x gcd ( x , 12 ) x 2 mod 12 deg ( x ) 2,10 2 4 7 3,9 3 9 6 4,8 4 4 5 5,7 1 1 8 6 6 0 4 Table 4. The vertex degrees of ∅ 0 ( Z 30 ). x g = gcd ( x , 12 ) x 2 mod 12 deg ( x ) 2,28 2 4 25 3,27 3 9 24 4,26 2 16 25 5,25 5 25 22 6,24 6 6 21 7,23 1 19 24 8,22 2 4 25 9,21 3 21 24 10,20 10 10 17 11,19 1 1 26 12,18 6 24 21 13,17 1 19 24 14,16 2 16 25 15 15 15 12 6. Topological graph indices This study calculates the essential topological indices, including the Zagreb and Randić, 15 indices, to assess connectedness, symmetry, and complexity. These calculations illustrate the intricate relationship between the ring-theoretic features and graph invariants. 1. First Zagreb Index: For any non-zero ideal I of Z n , and according to Theorem 5.2 , we obtain M 1 ( ∅ I ( Z n ) ) = ∑ x ∈ V ( ∅ I ( Zn ) ) ( dex ( x ) ) 2 = { ∑ x ∈ V ( ∅ I ( Z n ) ) ( n − n d − 3 ) 2 if g = 1 ∑ x ∈ V ( ∅ I ( Z n ) ) ( n − n g − 2 ) 2 if g > 1 If I = 0 , then according to Theorem 5.6 , we have ∑ x ∈ V ( ∅ 0 ( Z n ) ) ( dex ( x ) ) 2 = { ∑ x ∈ V ( ∅ 0 ( Z n ) ) ( n − 4 ) 2 if g = 1 ∑ x ∈ V ( ∅ 0 ( Z n ) ) ( n − g − 3 ) 2 if x 2 ≠ 0 ∑ x ∈ V ( ∅ 0 ( Z n ) ) ( n − g − 2 ) 2 if x 2 = 0 2. Second Zagreb Index: For any non-zero ideal I of Z n , let x , y ∈ V ( ∅ I ( Z n ) ) , let g x = gcd ( x , d ) , g y = gcd ( y , d ) and according to Theorem 5.2 , we obtain M 2 ( ∅ I ( Z n ) ) = ∑ xy ∈ E ( ∅ I ( Z n ) deg ( x ) deg ( y ) = { ∑ xy ∈ E ( ∅ I ( Z n ) ( n − n d − 3 ) 2 if g x = 1 , g y = 1 ∑ xy ∈ E ( ∅ I ( Z n ) ( n − n d − 3 ) ( n − n g y − 2 ) if g x = 1 , g y > 1 ∑ xy ∈ E ( ∅ I ( Z n ) ( n − n g x − 2 ) ( n − n d − 3 ) if g y = 1 , g x > 1 ∑ xy ∈ E ( ∅ I ( Z n ) ( n − n g x − 2 ) ( n − n g y − 2 ) if g y > 1 , g x > 1 3. Randić Index R ( ∅ I ( Z n ) ) = ∑ x y ∈ E ( ∅ I ( Z n ) 1 deg ( x ) deg ( y ) = { ∑ x y ∈ E ( ∅ I ( Z n ) 1 ( n − n d − 3 ) if g x = 1 , g y = 1 ∑ x y ∈ E ( ∅ I ( Z n ) 1 ( n − n d − 3 ) ( n − n g y − 2 ) if g x = 1 , g y > 1 ∑ x y ∈ E ( ∅ I ( Z n ) 1 ( n − n g x − 2 ) ( n − n d − 3 ) if g y = 1 , g x > 1 ∑ x y ∈ E ( ∅ I ( Z n ) 1 ( n − n g x − 2 ) ( n − n g y − 2 ) if g y > 1 , g x > 1 7. Conclusion In this study, we examined the ideal-based non-zero divisor graph ∅ I ( Z n ) , emphasizing the relationship between the algebraic structure of Z n and the combinatorial characteristics of its corresponding graph. We developed general formulas for vertex degrees, cut sets, and domination parameters as well as the conditions for graph connectedness. Topological degree-based indices were calculated to measure symmetry and structural complexity. The results show that ∅ I ( Z n ) is a useful way to observe and study ideal-related characteristics in modular rings, which connects algebraic and graph-theoretical points of view. Data availability No data are associated with this article. References 1. Anderson DF, Badawi A: The Zero-Divisor Graph of a Commutative Semigroup: A Survey. Groups, Modules, and Model Theory - Surveys and Recent Developments. Springer International Publishing; 2017; pp. 23–39. Publisher Full Text 2. Anderson DF, Axtell MC, Stickles JA: Zero-divisor graphs in commutative rings. Commutative Algebra: Noetherian and Non-Noetherian Perspectives. New York: Springer; 2011; pp. 23–45. Publisher Full Text 3. Anderson DD, Naseer M: Beck’s coloring of a commutative ring. J. Algebra. Aug. 1993; 159 (2): 500–514. Publisher Full Text 4. Livingston PS: Structure in Zero-Divisor Graphs of Commutative Rings.1997. Reference Source 5. Mcclurkin GE: Generalizations and Variations of the Zero-Divisor Graph. Reference Source 6. Dissertations D, Patrick Redmond S, Patrick S: Generalizations of the Zero-Divisor Graph of a Ring.2001. Reference Source 7. Kumari ML, Pandiselvi L, Palani K: Quotient Energy of Zero Divisor Graphs and Identity Graphs. Baghdad Sci. J. 2021; 20 (1): 277–282. Publisher Full Text 8. Redmond SP: An ideal-based zero-divisor graph of a commutative ring. Commun. Algebra. Sep. 2003; 31 (9): 4425–4443. Publisher Full Text 9. Kadem S, Aubad A, Majeed AH: The non-zero divisor graph of a ring. Ital. J. Pure Appl. Math. N. 2020; 43 (2020): 975–983. 10. Majeed AA, Aubad AA, Kadem S: The Non-Zero Divisor Graph of Prime Ring. Iraqi J. Sci. Jul. 2025; 2902–2908. Publisher Full Text 11. Sadiq FA, Ramadhan HA, Aubad AA: Some results on the non-zero divisor graphs of Zn. J. Discret. Math. Sci. Cryptogr. Jun. 2025; 28 (4-B): 1301–1307. Publisher Full Text 12. Gross JL, Yellen J, Anderson M: GRAPH THEORY AND app.2019. 13. Burton DM: Elementary number theory. McGraw-Hill; 2011. 14. Coté B, Ewing C, Huhn M, et al. : Cut-sets and Cut-vertices in the Zero-Divisor Graph of ∏Zni.2010. 15. Ghorbani M, Hosseinzade MA: A new version of Zagreb indices. Filomat. 2012; 26 (1): 93–100. Publisher Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 09 Jan 2026 ADD YOUR COMMENT Comment Author details Author details 1 Department of Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, Iraq 2 Department of Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, Iraq Sameer Kadem Roles: Conceptualization, Data Curation, Formal Analysis, Funding Acquisition, Methodology, Resources, Writing – Original Draft Preparation, Writing – Review & Editing Ali Abd Aubad Roles: Conceptualization, Investigation, Project Administration, Supervision Competing interests No competing interests were disclosed. Grant information This research was partially supported by the 4th Annual International Conference on Information and Sciences (AICIS’25), organized by the University of Fallujah. Further details about the conference can be found at: https://edas.info/index.php?c=34198 No additional grants or financial support were received from public, commercial, or not-for-profit funding agencies for the research, authorship, or publication of this article. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Article Versions (1) version 1 Published: 09 Jan 2026, 15:44 https://doi.org/10.12688/f1000research.172788.1 Copyright © 2026 Kadem S and Abd Aubad A. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. 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Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 09 Jan 2026 Views 0 Cite How to cite this report: Al-Sharqi F. Reviewer Report For: A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.5256/f1000research.190544.r449777 ) The direct URL for this report is: https://f1000research.com/articles/15-44/v1#referee-response-449777 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 10 Feb 2026 Faisal Al-Sharqi , University of Anbar, Ramadi, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.190544.r449777 Review on A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Zn Overall Assessment This study offers a substantial and systematic exploration of ideal-based non-zero divisor graphs over ∅��(Zn). Its findings—particularly regarding ... Continue reading READ ALL Review on A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Zn Overall Assessment This study offers a substantial and systematic exploration of ideal-based non-zero divisor graphs over ∅��(Zn). Its findings—particularly regarding continuity, the nature of segment sets, and the derivation of degree formulas—strengthen the connection between ring theoretical constructs and their graphing representations. Technical Quality The reasoning is robust and precise. All claims are supported by a logical sequence of results and carefully chosen examples. The work demonstrates extensive expertise in both algebra and graph theory. Presentation and Sequence The paper is clearly structured, guiding the reader smoothly from basic definitions to key theorems and their applications. Visual aids and tables are effectively used to illustrate key graph properties, making the material more accessible. Placement of the Research in Current Studies This work fits seamlessly into contemporary algebraic graph theory. By establishing general formulas, conditions, and indicators, this research provides a framework and useful tools for future studies on pattern loops and their graphical constants. Final Decision The manuscript is clearly written, mathematically accurate, and makes valuable contributions to the field. I support its acceptance for indexing, as it adheres to the conference's scientific standards and will be of interest to researchers in both algebra and graph theory. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Algebra and its applications I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Al-Sharqi F. Reviewer Report For: A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.5256/f1000research.190544.r449777 ) The direct URL for this report is: https://f1000research.com/articles/15-44/v1#referee-response-449777 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 25 Feb 2026 Ali Aubad , Department of Mathematics, University of Baghdad, Baghdad, Iraq 25 Feb 2026 Author Response Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero ... Continue reading Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero Divisor Graphs Associated with Zn." We are pleased that you recognised the significance of the connections we established between graph theory and ring theory. We sincerely value your insightful comments on the technical quality and presentation of the work. It is reassuring to see the soundness of our reasoning and the clarity of the research's organization, which facilitates the reader's understanding of the results. We are delighted that you have accepted the manuscript and considered it a valuable contribution to the field of algebraic graph theory. Thank you again for your time and expertise. Sincerely, The Authors Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero Divisor Graphs Associated with Zn." We are pleased that you recognised the significance of the connections we established between graph theory and ring theory. We sincerely value your insightful comments on the technical quality and presentation of the work. It is reassuring to see the soundness of our reasoning and the clarity of the research's organization, which facilitates the reader's understanding of the results. We are delighted that you have accepted the manuscript and considered it a valuable contribution to the field of algebraic graph theory. Thank you again for your time and expertise. Sincerely, The Authors Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 25 Feb 2026 Ali Aubad , Department of Mathematics, University of Baghdad, Baghdad, Iraq 25 Feb 2026 Author Response Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero ... Continue reading Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero Divisor Graphs Associated with Zn." We are pleased that you recognised the significance of the connections we established between graph theory and ring theory. We sincerely value your insightful comments on the technical quality and presentation of the work. It is reassuring to see the soundness of our reasoning and the clarity of the research's organization, which facilitates the reader's understanding of the results. We are delighted that you have accepted the manuscript and considered it a valuable contribution to the field of algebraic graph theory. Thank you again for your time and expertise. Sincerely, The Authors Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero Divisor Graphs Associated with Zn." We are pleased that you recognised the significance of the connections we established between graph theory and ring theory. We sincerely value your insightful comments on the technical quality and presentation of the work. It is reassuring to see the soundness of our reasoning and the clarity of the research's organization, which facilitates the reader's understanding of the results. We are delighted that you have accepted the manuscript and considered it a valuable contribution to the field of algebraic graph theory. Thank you again for your time and expertise. Sincerely, The Authors Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Arif NE and Majeed AA. Reviewer Report For: A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.5256/f1000research.190544.r449782 ) The direct URL for this report is: https://f1000research.com/articles/15-44/v1#referee-response-449782 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 27 Jan 2026 Nabeel Ezzulddin Arif , Mathematical Sciences, Tikrit University, Tikrit, Saladin, Iraq Amir A. Majeed , Mathematical Sciences, Sulaimani Polytechnic University (Ringgold ID: 467127), Sulaymaniyah, Kurdistan Region, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.190544.r449782 Review Report: A Study on the Structure of Ideal-Based Non- Zero Divisor Graphs Associated with Zn This manuscript presents a well-structured and mathematically rigorous study of non-zero divisor schemes and their variants ideal-based in the Z n ring. The ... Continue reading READ ALL Review Report: A Study on the Structure of Ideal-Based Non- Zero Divisor Graphs Associated with Zn This manuscript presents a well-structured and mathematically rigorous study of non-zero divisor schemes and their variants ideal-based in the Z n ring. The results are presented clearly, properly contextualized within the existing literature, and make a valuable contribution to the field. The manuscript's logical flow is sound, and the proofs are valid and thoroughly explained. It is recommended that this paper be indexed after the following minor revisions, which mainly concern symbolization, formatting, and presentation. Required Revisions: 1 - Lemma 3.3 : The formula need to rewrite because the largest ideal in Zn is generated by the smallest prime divisor , not via this formula. 2- Abstract : Justify the text alignment. 3- Figures : Place each figure in the text immediately after its first mention if possible . 4- Graph Notation ( Microsoft Equation Editor ): All mathematical symbols, especially those for graphs, should be formatted using Microsoft Equation Editor for consistency and professional presentation. No further mathematical review is required after these editorial changes. The paper is suitable for publication after review. Best regards, Prof. Dr. Nabeel E. Arif Dep. of Mathematics, College of Computer Science and Mathematics, Tikrit University, Iraq. HP: +9647701737803 Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly Competing Interests: No competing interests were disclosed. Reviewer Expertise: Graph Theory, Abstract Algebra, Topoligiacl spaces We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Arif NE and Majeed AA. Reviewer Report For: A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.5256/f1000research.190544.r449782 ) The direct URL for this report is: https://f1000research.com/articles/15-44/v1#referee-response-449782 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 09 Jan 2026 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 Version 1 09 Jan 26 read read Nabeel Ezzulddin Arif , Tikrit University, Tikrit, Iraq Amir A. Majeed , Sulaimani Polytechnic University (Ringgold ID: 467127), Sulaymaniyah, Iraq Faisal Al-Sharqi , University of Anbar, Ramadi, Iraq Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Al-Sharqi F. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 10 Feb 2026 | for Version 1 Faisal Al-Sharqi , University of Anbar, Ramadi, Iraq 0 Views copyright © 2026 Al-Sharqi F. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Review on A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Zn Overall Assessment This study offers a substantial and systematic exploration of ideal-based non-zero divisor graphs over ∅��(Zn). Its findings—particularly regarding continuity, the nature of segment sets, and the derivation of degree formulas—strengthen the connection between ring theoretical constructs and their graphing representations. Technical Quality The reasoning is robust and precise. All claims are supported by a logical sequence of results and carefully chosen examples. The work demonstrates extensive expertise in both algebra and graph theory. Presentation and Sequence The paper is clearly structured, guiding the reader smoothly from basic definitions to key theorems and their applications. Visual aids and tables are effectively used to illustrate key graph properties, making the material more accessible. Placement of the Research in Current Studies This work fits seamlessly into contemporary algebraic graph theory. By establishing general formulas, conditions, and indicators, this research provides a framework and useful tools for future studies on pattern loops and their graphical constants. Final Decision The manuscript is clearly written, mathematically accurate, and makes valuable contributions to the field. I support its acceptance for indexing, as it adheres to the conference's scientific standards and will be of interest to researchers in both algebra and graph theory. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Algebra and its applications I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (1) Author Response 25 Feb 2026 Ali Aubad, Department of Mathematics, University of Baghdad, Baghdad, Iraq Dear Reviewer, We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled "Ideal-Based Non-Zero Divisor Graphs Associated with Zn." We are pleased that you recognised the significance of the connections we established between graph theory and ring theory. We sincerely value your insightful comments on the technical quality and presentation of the work. It is reassuring to see the soundness of our reasoning and the clarity of the research's organization, which facilitates the reader's understanding of the results. We are delighted that you have accepted the manuscript and considered it a valuable contribution to the field of algebraic graph theory. Thank you again for your time and expertise. Sincerely, The Authors View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Al-Sharqi F. Peer Review Report For: A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.5256/f1000research.190544.r449777) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-44/v1#referee-response-449777 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Arif N et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 27 Jan 2026 | for Version 1 Nabeel Ezzulddin Arif , Mathematical Sciences, Tikrit University, Tikrit, Saladin, Iraq Amir A. Majeed , Mathematical Sciences, Sulaimani Polytechnic University (Ringgold ID: 467127), Sulaymaniyah, Kurdistan Region, Iraq 0 Views copyright © 2026 Arif N et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Review Report: A Study on the Structure of Ideal-Based Non- Zero Divisor Graphs Associated with Zn This manuscript presents a well-structured and mathematically rigorous study of non-zero divisor schemes and their variants ideal-based in the Z n ring. The results are presented clearly, properly contextualized within the existing literature, and make a valuable contribution to the field. The manuscript's logical flow is sound, and the proofs are valid and thoroughly explained. It is recommended that this paper be indexed after the following minor revisions, which mainly concern symbolization, formatting, and presentation. Required Revisions: 1 - Lemma 3.3 : The formula need to rewrite because the largest ideal in Zn is generated by the smallest prime divisor , not via this formula. 2- Abstract : Justify the text alignment. 3- Figures : Place each figure in the text immediately after its first mention if possible . 4- Graph Notation ( Microsoft Equation Editor ): All mathematical symbols, especially those for graphs, should be formatted using Microsoft Equation Editor for consistency and professional presentation. No further mathematical review is required after these editorial changes. The paper is suitable for publication after review. Best regards, Prof. Dr. Nabeel E. Arif Dep. of Mathematics, College of Computer Science and Mathematics, Tikrit University, Iraq. HP: +9647701737803 Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Partly Competing Interests No competing interests were disclosed. Reviewer Expertise Graph Theory, Abstract Algebra, Topoligiacl spaces We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Arif NE and Majeed AA. Peer Review Report For: A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n [version 1; peer review: 2 approved] . F1000Research 2026, 15 :44 ( https://doi.org/10.5256/f1000research.190544.r449782) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-44/v1#referee-response-449782 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. 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