A note on eigenvalues of zero divisor graphs associated with commutative rings
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| Format: | Preprint |
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2024
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| _version_ | 1866910297752600576 |
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| author | Rather, Bilal Ahmad |
| author_facet | Rather, Bilal Ahmad |
| contents | For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $Γ(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(Γ(R))$ are adjacent if and only if $x\cdot y=0.$ In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \cite{johnson}]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring $\mathbb{Z}_{p}[x]/\langle x^{4} \rangle.$ We show that for any prime $p$, $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is non-hyperenergetic and for prime $p\geq 3$, $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is hypoenergetic. We give a formulae for the topological indices of $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cević conjecture \cite{hansen}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_02554 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on eigenvalues of zero divisor graphs associated with commutative rings Rather, Bilal Ahmad Combinatorics Discrete Mathematics Rings and Algebras Spectral Theory 05C50, 05C25, 15A18, 13A70 For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $Γ(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(Γ(R))$ are adjacent if and only if $x\cdot y=0.$ In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \cite{johnson}]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring $\mathbb{Z}_{p}[x]/\langle x^{4} \rangle.$ We show that for any prime $p$, $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is non-hyperenergetic and for prime $p\geq 3$, $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is hypoenergetic. We give a formulae for the topological indices of $Γ(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cević conjecture \cite{hansen}. |
| title | A note on eigenvalues of zero divisor graphs associated with commutative rings |
| topic | Combinatorics Discrete Mathematics Rings and Algebras Spectral Theory 05C50, 05C25, 15A18, 13A70 |
| url | https://arxiv.org/abs/2401.02554 |