2024/06/02 by Surbhi, Geetha Venkataraman, Venkataraman, Geetha
Mathematics · #20D60 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2406.00710
openalex publication_date 2024/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The power graph of a group G is a graph with vertex set G, in which two vertices are adjacent if one is some power of the other. In the commuting graph, with G as the vertex set, two vertices are joined by an edge if they commute in G. The enhanced power graph of a group G is a graph with vertex set G and an edge joining two vertices x and y if ⟨ x,y⟩ is cyclic. In this paper, we answer a question posed by P. J. Cameron, namely, if there exist groups G and H such that the power graph of G is isomorphic to the commuting graph of H. We show that the answer is yes if G is the generalised quaternion group and H is the dihedral group. We also show that the enhanced power graph of the dicyclic group is isomorphic to the commuting graph of the dihedral group.