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

Let [Formula: see text] be a commutative ring with unity. The essential ideal graph [Formula: see text] of [Formula: see text] is a graph in which the vertex set comprises of set of all nonzero proper ideals of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] 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], [Formula: see text] 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 [Formula: see text] for which these bounds are attained.