Adjacency Spectrum and Wiener Index of the Essential Ideal Graph of a Finite Commutative Ring ${\mathbb{Z}}_n$

Abstract

Let $R$ be a commutative ring with unity. The essential ideal graph ${\mathcal{E}}_R$ of $R$, is a graph with a vertex set consisting of all nonzero proper ideals of \textit{R} and two vertices $I$ and $K$ are adjacent if and only if $I+K$ is an essential ideal. In this paper, we study the adjacency spectrum of the essential ideal graph of the finite commutative ring ${\mathbb{Z}}_n$, for $n=\{p^m,p^{m_1}{q}^{m_2}\}$, where $p,q$ are distinct primes, and $m,m_1,m_2\in \mathbb{N}$. We show that $0$ is an eigenvalue of the adjacency matrix of ${\mathcal{E}}_{{\mathbb{Z}}_n}$ if and only if either $n=p^2$ or $n$ is not a product of distinct primes. We also determine all the eigenvalues of the adjacency matrix of ${\mathcal{E}}_{{\mathbb{Z}}_n}$ whenever $n$ is a product of three or four distinct primes. Moreover, we calculate the topological indices, namely the Wiener index and hyper-Wiener index of the essential ideal graph of ${\mathbb{Z}}_n$ for different forms of $n$

18 pages, 1 figure