Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring
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| Format: | Preprint |
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2026
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| _version_ | 1866913161724035072 |
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| author | Shylla, Hampher Mawiong, Sainkupar Mn Kharbhih, John Paul Jala |
| author_facet | Shylla, Hampher Mawiong, Sainkupar Mn Kharbhih, John Paul Jala |
| contents | For a commutative ring $R$ with identity, the \emph{weakly zero-divisor graph} $WΓ(R)$ 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 $WΓ(Z_n)$ has been determined by Shariq, Mathil, and Kumar, who also established that $WΓ(Z_n)$ is Laplacian integral. Building on the structural description of $WΓ(R)$ 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 $WΓ(R)$ 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 $WΓ(R)$, 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 $WΓ(R)$, and recover the Laplacian spectrum of $WΓ(Z_n)$ in compact form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24640 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring Shylla, Hampher Mawiong, Sainkupar Mn Kharbhih, John Paul Jala Combinatorics For a commutative ring $R$ with identity, the \emph{weakly zero-divisor graph} $WΓ(R)$ 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 $WΓ(Z_n)$ has been determined by Shariq, Mathil, and Kumar, who also established that $WΓ(Z_n)$ is Laplacian integral. Building on the structural description of $WΓ(R)$ 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 $WΓ(R)$ 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 $WΓ(R)$, 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 $WΓ(R)$, and recover the Laplacian spectrum of $WΓ(Z_n)$ in compact form. |
| title | Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.24640 |