Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring

Abstract

For a commutative ring $R$ with identity, the \emph{weakly zero-divisor graph} $W\Gamma(R)$ has vertex set $Z(R)^{\ast}$, with distinct vertices $x$ and $y$ adjacent whenever there exist nonzero $r\in{\rm Ann}(x)$ and $s\in{\rm Ann}(y)$ with $rs=0$. The Laplacian spectrum of $W\Gamma(Z_n)$ has been determined by Shariq, Mathil, and Kumar, who also established that $W\Gamma(Z_n)$ is Laplacian integral. Building on the structural description of $W\Gamma(R)$ due to Nikmehr, Azadi, and Nikandish, we extend the Laplacian spectrum and integrality results from $Z_n$ to \emph{every} finite commutative ring $R$: we restate $W\Gamma(R)$ in unified form as a complete multipartite graph whose parts are made explicit by the local-ring decomposition of $R$, compute the full Laplacian spectrum in closed form, prove Laplacian integrality of $W\Gamma(R)$, and give a sharp bound on the number of distinct Laplacian eigenvalues. As consequences we obtain explicit formulas for the algebraic connectivity and number of spanning trees of $W\Gamma(R)$, and recover the Laplacian spectrum of $W\Gamma(Z_n)$ in compact form.