Laplacian spectrum of weakly zero-divisor graph of the ring $\mathbb{Z}_{n}$

TL;DR

This paper investigates the Laplacian spectrum of the weakly zero-divisor graph of the ring rac{n}{}, proving it is Laplacian integral for all positive integers n, extending known results from prime power rings.

Contribution

It determines the Laplacian spectrum of the weakly zero-divisor graph for rac{n}{} and establishes its Laplacian integrality for all n, generalizing prior results.

Findings

The Laplacian spectrum of WGamma(rac{n}{}) is explicitly obtained.

WGamma(rac{n}{}) is Laplacian integral for all positive integers n.

The results extend known spectral properties from prime power rings to arbitrary n.

Abstract

Let $R$ be a commutative ring with unity. The weakly zero-divisor graph $W\Gamma(R)$ of the ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two vertices $x$, $y$ are adjacent if and only if there exists $r\in {\rm ann}(x)$ and $s \in {\rm ann}(y)$ such that $rs =0$. The zero-divisor graph of a ring is a spanning subgraph of the weakly zero-divisor graph. It is known that the zero-divisor graph of the ring $\mathbb{Z}_{{p^t}}$, where $p$ is a prime, is the Laplacian integral. In this paper, we obtain the Laplacian spectrum of the weakly zero-divisor graph $W\Gamma(\mathbb{Z}_{n})$ of the ring $\mathbb{Z}_{n}$ and show that $W\Gamma(\mathbb{Z}_{n})$ is Laplacian integral for arbitrary $n$.