  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 $n$
