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 $r\in {\rm ann}(x)$ 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 $n$.