A novel approach to explore common prime divisor graphs and their degree based topological descriptor.

For the construction of a common prime divisor graph, we consider an integer [Formula: see text] with its prime factorization, where [Formula: see text] are distinct primes and [Formula: see text] are fixed positive integers. Every divisor of the integer [Formula: see text] has the form [Formula: see text], with [Formula: see text]. There are [Formula: see text] distinct divisors of integer [Formula: see text]. Let [Formula: see text] be the collection of all positive divisors of [Formula: see text] other than integer 1. Then we can define a simple graph on the set of divisors [Formula: see text] of [Formula: see text], called a common prime divisor graph [Formula: see text] with [Formula: see text] as the vertex set, and we insert an edge between two distinct divisors x and y of [Formula: see text] if the [Formula: see text]. In this article, we will introduce and discuss some basic properties of common prime divisor graphs and we will compute some indices of symmetries associated with a class of such graphs. This study will open a new domain of graphs to investigate their invariant and to explore such indices on the different classes of common prime divisor graphs.