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 ∈ 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 Γ 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. © 2024 Forum-Editrice Universitaria Udinese SRL. All rights reserved.