Abstract:For a commutative ring $R$ with identity, the \emph{weakly zero-divisor graph} $\WG(R)$ has vertex set $\Z(R)^{\ast}$, with distinct vertices $x$ and $y$ adjacent whenever there exist nonzero $r\in\Ann(x)$ and $s\in\Ann(y)$ with $rs=0$. The Laplacian spectrum of $\WG(\Z_n)$ has been determined by Shariq, Mathil, and Kumar, who also established that $\WG(\Z_n)$ is Laplacian integral. Building on the structural description of $\WG(R)$ 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 $\WG(R)$ 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 $\WG(R)$, 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 $\WG(R)$, and recover the Laplacian spectrum of $\WG(\Z_n)$ in compact form.
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.24640 [math.CO] |
| (or arXiv:2605.24640v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24640 arXiv-issued DOI via DataCite (pending registration) |
From: Sainkupar Marwein Mawiong [view email]
[v1]
Sat, 23 May 2026 16:11:54 UTC (17 KB)
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