Between the enhanced power graph and the commuting graph. Issue 2 (8th August 2022)

• Record Type: Journal Article
• Authors: Cameron, Peter J.; Kuzma, Bojan
• Is Part Of: Journal of graph theory. Volume 102:Issue 2(2023)
• Page Start: 295
• Page End: 303
• Publication Date: 2022-08-08
• Subjects: Bogomolov multiplier -- central extension -- commuting graph -- enhanced power graph -- Schur multiplier; Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.22871
• Languages: English
• ISSNs: 0364-9024

Abstract

Abstract: The purpose of this note is to define a graph whose vertex set is a finite group G $G$, whose edge set is contained in that of the commuting graph of G $G$ and contains the enhanced power graph of G $G$ . We call this graph the deep commuting graph of G $G$ . Two elements of G $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of G $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of G $G$ act as automorphisms of the deep commuting graph.