On divisor labeling of co-prime order graphs of finite groups

The co -prime order graph of a finite group G is an undirected graph whose vertex set is G and two distinct vertices u, v is an element of G are adjacent if gcd ( o ( u ) , o ( v )) = 1 or a prime number. Labeling a graph is the process of assigning integers to its vertices and/or edges subject to certain conditions. In other words, vertex (edge) labeling is a function of the set of vertices (edges) to a set of labels (generally integers). A graph Gamma is a divisor graph if all its vertices can be labeled with positive integers such that two distinct vertices x and y are adjacent if and only if x | y or y | x . This paper focuses on some conditions under which the co -prime order graphs of finite groups, especially abelian groups and permutation groups, are divisor graphs.