Relative g-noncommuting graph of finite groups

Abstract

Let G be a finite group. For a fixed element g in G and a given subgroup H of G , the relative g -noncommuting graph of G is a simple undirected graph whose vertex set is G and two vertices x and y are adjacent if x ∈ H or y ∈ H and [ x , y ]≠ g , g −1 . We denote this graph by Γ H , G g . In this paper, we obtain computing formulae for degree of any vertex in Γ H , G g and characterize whether Γ H , G g is a tree, star graph, lollipop or a complete graph together with some properties of Γ H , G g involving isomorphism of graphs. We also present certain relations between the number of edges in Γ H , G g and certain generalized commuting probabilities of G which give some computing formulae for the number of edges in Γ H , G g . Finally, we conclude this paper by deriving some bounds for the number of edges in Γ H , G g .