SEMINAR: Groups and Combinatorics Seminar, On The Non-commuting Graph

Abstract:

In this talk, we will consider the non-commuting graph of a non-abelian finite group G; its vertex set is the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if they do not commute together. Actually, we study some properties of the non-commuting graph such as connectivity, regularity, etc., and we show that, for many groups G, if H is a group which has the same non-commuting graph of G, then they have the same order. We determine the structure of any finite non-abelian group G (up to isomorphism) for which its non-commuting graph is a complete multipartite graph. We also show that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph.