Which coprime graphs are divisor graphs?

Abstract: For a finite group $G$, the coprime graph $\Gamma(G)$ of $G$ is a graph with vertex set $G$, in which two distinct vertices $a$ and $b$ are adjacent if the order of $a$ and the order of $b$ are coprime. In this paper, we first give a characterization for which generalized lexicographic products are divisor graphs. As applications, we show that every of power graph, reduced power graph and order graph is a divisor graph, which also implies the main result in [N. Takshak, A. Sehgal, A. Malik, Power graph of a finite group is always divisor graph, Asian-Eur. J. Math. 16 (2023), ID: 2250236]. Then, we prove that the coprime graph of a group is a generalized lexicographic product, and give two characterizations for which coprime graphs are divisor graphs. We also describe the groups $G$ with $|\pi(G)|\le 4$, whose coprime graph is a divisor graph. Finally, we classify the finite groups $G$ so that $\Gamma(G)$ is a divisor graph 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.