Adjacency Spectrum and Wiener Index of the Essential Ideal Graph of a Finite Commutative Ring \(\mathbb{Z}_{n}\)

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 \(n\)