2025/01/24 by Ma, Xuanlong, Zhai, Liangliang, Gao, Nan
#05C25 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2501.14339
For a finite group G, the coprime graph Γ(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 |π(G)|≤ 4, whose coprime graph is a divisor graph. Finally, we classify the finite groups G so that Γ(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.