Non-Commuting Graphs of AC-Groups are End-Regular

Abstract

A graph is called End-regular if its endomorphism monoid is regular. Which graphs are End-regular? This is an open question and difficult to obtain a general answer. In the present paper, we investigate the End-regularity of graphs obtained by adding or deleting vertices from End-regular graphs. As an application, we show that the non-commuting graphs of \(AC\)-groups are End-regular.