Anita Lande(MES Abasaheb Garware College (Autonomous), Affiliated to Savitribai Phule Pune University)0009-0005-8542-6290
Geetali Bhavar(Department of Mathematics, MES Abasaheb Garware College (Autonomous), Affiliated to Savitribai Phule Pune University)0009-0005-2794-3046
In this paper, we determine the Wiener and Harary indices of $\Gamma'(\mathbb Z_{n})$ for $n=pq,p^2q,p^2q^2$, where $p, q$ are distinct primes.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $R$ be a commutative ring, and let $Z^*(R)$ denote the set of all non-zero zero-divisors in $R$.
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Key Findings & Data:The generalized zero-divisor graph of $R$, denoted by $\Gamma'(R)$, is a simple (undirected) graph with the vertex set $Z^*(R)$, and two distinct vertices $x$ and $y$ are adjacent in $\Gamma'(R)$ if and only if $x^ny=0$ or $y^nx=0$, for some positive integer $n$.
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Conclusion & Impact:In this paper, we determine the Wiener and Harary indices of $\Gamma'(\mathbb Z_{n})$ for $n=pq,p^2q,p^2q^2$, where $p, q$ are distinct primes.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Zero-divisor graph · Wiener index · Harary index
Abstract
Let $R$ be a commutative ring, and let $Z^*(R)$ denote the set of all non-zero zero-divisors in $R$. The generalized zero-divisor graph of $R$, denoted by $\Gamma'(R)$, is a simple (undirected) graph with the vertex set $Z^*(R)$, and two distinct vertices $x$ and $y$ are adjacent in $\Gamma'(R)$ if and only if $x^ny=0$ or $y^nx=0$, for some positive integer $n$. The Wiener and Harary indices are topological indices widely used in chemistry to analyze the structures of molecules. In this paper, we determine the Wiener and Harary indices of $\Gamma'(\mathbb Z_{n})$ for $n=pq,p^2q,p^2q^2$, where $p, q$ are distinct primes.
Khairnar, A., Lande, A., Bhavar, G., Phatangare, N. (2026). WIENER AND HARARY INDICES OF THE GENERALIZED ZERO-DIVISOR GRAPH. TWMS Journal of Applied and Engineering Mathematics, Volume: 16 Issue: 10. https://ajindex.com/articles/1604190
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