On the structure and spectra of an induced subgraph of essential ideal graph of $\mathbb{Z}_{n}$

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.