  Let $R$ be a commutative ring with unity. The essential ideal graph
$\mathcal{E}_R$ of $R$ is a graph in which the vertex set comprises of set 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 discuss the
structure of an induced subgraph of the essential ideal graph of the ring
$\mathbb{Z}_{n}$ as a $\mathscr{G}$-generalized join graph and thereby
completely determine the structure of $\mathcal{E}_{\mathbb{Z}_{n}}$. Also, we
prove a characterization of $\mathcal{E}_{\mathbb{Z}_{n}}$ to be Laplacian
integral in terms of the vertex-weighted Laplacian matrix of annihilating ideal
graph of $\mathbb{Z}_{n}$ for $n= \prod_{i=1}^k p_i$. Further, we discuss the
eigenvalues of various matrices like adjacency matrix, Laplacian matrix,
signless Laplacian matrix, and normalized Laplacian matrix of the induced
subgraph of the essential ideal graph of $\mathbb{Z}_{n}$. Finally, we obtain
the upper bounds of spectral radius and algebraic connectivity of
$\mathcal{E}_{\mathbb{Z}_{n}}$ and compute the values of $n$ for which these
bounds are attained.

Commutative Algebra;Combinatorics;Rings and Algebras