# Computation of Some Graph Energies of the Zero-Divisor Graph Associated‎~ ‎with the Commutative Ring $\mathbb{Z}_{p^{2}}[x]/\langle x^{2} \rangle$

Document Type : Research Paper

Authors

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India

Abstract

‎Let $\mathcal{R}$ be the commutative ring $\mathcal{R}=\mathbb{Z}_{p^2}[x]/\langle x^{2} \rangle$ with identity and ${Z^{*}}(\mathcal{R})$ be the set of all non-zero zero-divisors of $\mathcal{R}$‎. ‎Then‎, ‎$\Gamma(\mathcal{R})$ is said to be a zero-divisor graph if and only if $a \cdot b= 0$ where $a,b \in V(\Gamma(\mathcal{R})) = {Z^{*}}(\mathcal{R})$ and $(a,b) \in E(\Gamma(\mathcal{R}))$‎. ‎Let $\lambda_1,\lambda_2,\dots,\lambda_n$ be the eigenvalues of the adjacency matrix‎, ‎and let $\mu_1,\mu_2,\dots,\mu_n$ be the eigenvalues of the Laplacian matrix of $\Gamma(\mathcal{R})$‎. ‎Then %the energy of $\Gamma(\mathcal{R})$ is defined as the sum of the absolute values of the eigenvalues of the graph $\Gamma(\mathcal{R})$ and the Laplacian energy of $\Gamma(\mathcal{R})$ is the sum of the absolute deviations of its Laplacian matrix's eigenvalues of the graph $\Gamma(\mathcal{R})$‎. ‎In this paper‎,
‎we discuss the energy $\mathcal{E}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\lambda_{i}}$ and the Laplacian energy $\mathcal{LE}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\mu_{i}-\frac{2m}{n}}$ where $n$ and $m$ are the order and size of $\Gamma(\mathcal{R})$‎.

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Main Subjects

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