Universal adjacency spectrum of the looped zero divisor graph for a finite commutative ring with unity

Abstract

For a finite undirected looped graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. For a finite commutative ring [Formula: see text] with unity, the looped zero divisor graph [Formula: see text] is an undirected graph with the set of all nonzero zero divisors of [Formula: see text] as vertices and two vertices (not necessarily distinct) [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some structural properties of [Formula: see text] by defining an equivalence relation on its vertex set. Then we obtain the universal adjacency eigenpairs of [Formula: see text], and as a consequence several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree matrix of [Formula: see text] can be obtained in a unified way. Moreover, we get the structural properties and the universal adjacency eigenpairs of the looped zero divisor graph of a reduced ring in a simpler form.