On the graphical representation of an integer

For a given positive integer [Formula: see text], the prime-[Formula: see text] graph denoted by [Formula: see text] is defined on the vertex set comprising all positive divisors of [Formula: see text] greater than 1. An edge exists between two distinct vertices [Formula: see text] and [Formula: see text] if and only if their greatest common divisor, [Formula: see text], is a prime factor of [Formula: see text]. This study explores the fundamental structural characteristics of [Formula: see text] systematically. Key findings establish that the graph is always connected for any integer [Formula: see text], with a diameter of at most [Formula: see text] and a radius of [Formula: see text]. The paper provides characterizations and formulas for various graph invariants, including the clique number, chromatic number, girth, and vertex degrees, demonstrating their direct dependence on the prime factorization of [Formula: see text]. It is shown that graphs [Formula: see text] and [Formula: see text] are isomorphic if [Formula: see text] and [Formula: see text] share the same prime factorization structure irrespective of the prime factors. Furthermore, conditions for planarity are determined. The analysis also covers properties such as the independence number, covering number, density of a graph establishing a relation between them and prime factorization of [Formula: see text]. This research illuminates the deep interplay between the arithmetic properties of integers and the resulting topological features of their associated graphs.