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

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 $r\in \mathrm{ann}(x)$ and $s\in \mathrm{ann}(y)$ 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 $n$.

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