The adjacency spectrum and metric dimension of an induced subgraph of comaximal graph of ℤn

Abstract

Let R be a commutative ring with unity, and let [Formula: see text] denote the comaximal graph of R. The comaximal graph [Formula: see text] has vertex set as R, and any two distinct vertices x, y of [Formula: see text] are adjacent if [Formula: see text]. Let [Formula: see text] denote the induced subgraph of [Formula: see text] on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of [Formula: see text] are adjacent if [Formula: see text]. In this paper, we study the graphical structure as well the adjacency spectrum of [Formula: see text], where [Formula: see text] is a non-prime positive integer, and [Formula: see text] is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of [Formula: see text] are [Formula: see text] with multiplicity [Formula: see text], and remaining eigenvalues are contained in the spectrum of a symmetric [Formula: see text] matrix. We further calculate the rank and nullity of [Formula: see text]. We also determine all the eigenvalues of [Formula: see text] whenever [Formula: see text] is a bipartite graph. Finally, apart from determining certain structural properties of [Formula: see text], we conclude the paper by determining the metric dimension of [Formula: see text].