On Zero-Divisor Graph of the Ring $\frac{\mathbb{F}_p[u, v]}{\langle u^2,\, v^2, \, uv-vu\rangle}$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910231570677760 |
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| author | Annamalai, N. |
| author_facet | Annamalai, N. |
| contents | In this article, we study the zero-divisor graph of the commutative non-chain ring with identity $ \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where \(u^2 = 0\), \(v^2 = 0\), \(uv = vu\), and \(p\) is an odd prime. We determine several graph-theoretic properties of the associated zero-divisor graph \(Γ(R)\), including clique number, chromatic number, vertex connectivity, edge connectivity, diameter, and girth. In addition, we compute certain topological indices of \(Γ(R)\). Furthermore, we obtain the eigenvalues, energy, and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of \(Γ(R)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17847 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Zero-Divisor Graph of the Ring $\frac{\mathbb{F}_p[u, v]}{\langle u^2,\, v^2, \, uv-vu\rangle}$ Annamalai, N. Rings and Algebras Combinatorics In this article, we study the zero-divisor graph of the commutative non-chain ring with identity $ \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where \(u^2 = 0\), \(v^2 = 0\), \(uv = vu\), and \(p\) is an odd prime. We determine several graph-theoretic properties of the associated zero-divisor graph \(Γ(R)\), including clique number, chromatic number, vertex connectivity, edge connectivity, diameter, and girth. In addition, we compute certain topological indices of \(Γ(R)\). Furthermore, we obtain the eigenvalues, energy, and spectral radius of the adjacency matrix, the Laplacian matrix and the Eccentricity matrix of \(Γ(R)\). |
| title | On Zero-Divisor Graph of the Ring $\frac{\mathbb{F}_p[u, v]}{\langle u^2,\, v^2, \, uv-vu\rangle}$ |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2605.17847 |