Sombor index and eigenvalues of weakly zero-divisor graph of commutative rings

TL;DR

We study the Sombor index and spectrum of the weakly zero-divisor graph of the ring $\mathbb{Z}_n$. The approach uses a partition of the vertex set by the divisor-associated sets $\mathcal{A}_{d}$ and expresses $W\Gamma(\mathbb{Z}_n)$ as a generalized join, enabling closed-form expressions for $SO(W\Gamma(\mathbb{Z}_n))$ and the Sombor spectrum via equitable partitions. Key results include explicit formulas for $SO(W\Gamma(\mathbb{Z}_n))$ across several arithmetic forms of $n$ (including prime-power and mixed prime-power cases), exact eigenvalue structures with multiplicities tied to $\phi(n)$ and related totients, and derived lower bounds for Sombor energy. These findings extend the understanding of Sombor invariants on ring-based graph constructions and provide precise spectral data useful for algebraic graph theory and potential chemical applications.

Abstract

The weakly zero-divisor graph $WΓ(R)$ of a commutative ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two distinct vertices $x$, $y$ are adjacent if and only if there exist $w\in {\rm ann}(x)$ and $ z\in {\rm ann}(y)$ such that $wz =0$. In this paper, we determine the Sombor index for the weakly zero-divisor graph of the integers modulo ring $\mathbb{Z}_n$. Furthermore, we investigate the Sombor spectrum and establish bounds for the Sombor energy of the weakly zero-divisor graph of $\mathbb{Z}_n$.

Figures (1)

• Figure 1: The graph $W\Gamma(\mathbb{Z}_{18})$

Theorems & Definitions (21)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Corollary 2.5
• Lemma 2.6
• Example 2.7
• Theorem 2.8
• proof
• Corollary 2.9