On the structure and spectra of an induced subgraph of essential ideal graph of ℤn

Let R be a commutative ring with unity. The essential ideal graph Formula: see text 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 J are adjacent if and only if I + J is an essential ideal. In this paper, we discuss the structure of an induced subgraph of the essential ideal graph of the ring Formula: see text as a Formula: see text-generalized join graph and thereby completely determine the structure of Formula: see text. Also, we prove a characterization of Formula: see text to be Laplacian integral in terms of the vertex-weighted Laplacian matrix of annihilating ideal graph of Formula: see text for Formula: see text, p i are different prime numbers. Further, we estimate 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 Formula: see text. Finally, we obtain the upper bounds of spectral radius and algebraic connectivity of Formula: see text and compute the values of n for which these bounds are attained