Spectra of zero-divisor graphs of finite reduced rings

Abstract

Let [Formula: see text] be a finite reduced ring with [Formula: see text] maximal ideals [Formula: see text] and let [Formula: see text] be the zero-divisor graph associated to [Formula: see text] The class of rings [Formula: see text] contains the Boolean rings as a subclass. When [Formula: see text] for all [Formula: see text] where [Formula: see text] is a finite field, we associate two [Formula: see text] sized matrices [Formula: see text] and [Formula: see text] to the graph [Formula: see text] having combinatorial entries and use these matrices to determine the spectrum of this graph. More precisely, we show that every eigenvalue of [Formula: see text] and of [Formula: see text] is an eigenvalue of [Formula: see text] To do this, we give a recursive description of the adjacency matrix of this graph and also exhibit its equitable partition. This is used in computing the determinant, rank and nullity of the adjacency matrix. Further, we propose that the eigenvalues of [Formula: see text] [Formula: see text] and the eigenvalue [Formula: see text] exhaust all the eigenvalues of [Formula: see text]