On some structural properties of a divisor graph

A divisor graph G n 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 G n and G′ n . Formally, we determine the clique number, the independence number, the chromatic number and the automorphism group of the divisor graphs G n 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.

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