On finite groups whose coprime graph is a divisor graph

Abstract

In this paper, we first characterize which generalized lexicographic products are divisor graphs. As applications, we show that power graphs, reduced power graphs and order graphs are all divisor graphs, which also implies the main result in [Power graph of a finite group is always divisor graph, Asian-European Journal of Mathematics 16 (2023)]. We then show that, the coprime graph of a group is a generalized lexicographic product, and characterize which coprime graphs are divisor graphs. Finally, we classify the finite groups $G$ having at most four prime divisors, whose coprime graphs are divisor graphs, and we also classify the finite groups $G$ whose coprime graphs are divisor graphs, if $G$ is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, a direct product of two non-trivial groups, and a sporadic simple group.