2025/11/17 by Sumana Hatui, Hatui, Sumana, Sanjay Mukherjee +3
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #Interconnection Networks and Systems
paper · pdf · doi:10.48550/arxiv.2511.13303
openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
Let G be a finite group and let G be a Schur cover of G. The deep commuting graph ΔD(G) of G is a simple graph with vertex set G, where two distinct vertices are adjacent if their pre-images commute in G. The deep commuting graph of a finite group was first introduced in [P. J. Cameron and B. Kuzma, Between the enhanced power graph and the commuting graph, \it J. Graph Theory \bf 102 (2023), no. 2, 295--303], where the authors have shown that ΔD(G) is fixed irrespective of the choice of the Schur cover G. In this paper, we first prove that ΔD(G) is complete if and only if G is cyclic. Also, we classify finite simple groups, symmetric groups and alternating groups, for which ΔD(G) is perfect. In addition, explore several other properties of ΔD(G) like Eulerianess, universality and connectedness of reduced deep commuting graphs. Next, we classify the finite abelian groups for which deep commuting graphs coincide with enhance power graphs. We also characterize the dominant vertices for the deep commuting graphs of finite abelian groups and examine the connectedness of the associated reduced deep commuting graphs. These properties of the deep commuting graphs for the non abelian groups like symmetric groups, alternating groups, dihedral groups, generalized quaternion group and Heisenberg groups are also discussed.