On some structural properties of a divisor graph

A divisor graph Gn of a positive integer n is a simple graph with vertices as proper divisors of n, in which two distinct divisors are adjacent if and only if they are relatively prime. Also G′n is a graph with vertices as divisors of n except n, such that the two distinct vertices are adjacent if and only if their greatest common divisor is one. These graph play a fundamental role in the study of the comaximal graphs associated to commutative rings. We study the graph theoretic properties of both Gn and G′n. Formally, we determine the clique number, the independence number, the chromatic number and the automorphism group of the divisor graphs Gn and G′n. We also find the bounds for their metric dimension along with the characterization of graphs attaining them. Further, we discuss their resolving polynomial.