# Adjacency spectrum and Wiener index of essential ideal graph of a finite commutative ring

### Abstract

Let R be a commutative ring with unity. The essential ideal graph* E*_{R} of R, is a graph with a vertex set consisting of all nonzero proper ideals of 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 Z_{n,} for n = {p^{m}, p^{m1}q^{m2}}, where p, q are distinct primes, and m, m_{1} , m_{2} ε N. We show that 0 is an eigenvalue of the adjacency matrix of E_{Zn }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 E_{Zn }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 Z_{n}, for different forms of n.