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

## TL;DR

This paper introduces a new type of graph based on common prime divisors and explores its properties and symmetry indices.

## Contribution

The paper introduces common prime divisor graphs and computes symmetry indices for these graphs.

## Key findings

- Common prime divisor graphs are defined based on the divisors of an integer and their prime factors.
- Symmetry indices are calculated for a class of these graphs.
- The study opens a new area for exploring graph invariants in different classes of these graphs.

## Abstract

For the construction of a common prime divisor graph, we consider an integer ζ=∏i=1kpiγi≥2 with its prime factorization, where pi′s are distinct primes and γi′s are fixed positive integers. Every divisor of the integer ζ has the form x=∏i=1kpixi, with 0≤xi≤γi. There are ∏i=1k(γi+1) distinct divisors of integer ζ. Let D(ζ) be the collection of all positive divisors of ζ other than integer 1. Then we can define a simple graph on the set of divisors D(ζ) of ζ, called a common prime divisor graph
ℨ(ζ) with D(ζ) as the vertex set, and we insert an edge between two distinct divisors x and y of ζ if the gcd(x,y)=pi. 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.