Let R be a commutative ring with unity. The essential ideal graph ER 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 Zn, for n = {pm, pm1qm2}, where p, q are distinct primes, and m, m1 , m2 ε N. We show that 0 is an eigenvalue of the adjacency matrix of EZn if and only if either n = p2 or n is not a product of distinct primes. We also determine all the eigenvalues of the adjacency matrix of EZn 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 Zn, for different forms of n.