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

Abstract

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 $\mathcal{G}$-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.

21 pages, 2 figures