Relative g-noncommuting graph of finite groups - Monalisha Sharma; Rajat Kanti Nath (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2022)
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 ΓH, Gg. In this paper, we obtain computing formulae for degree of any vertex in ΓH, Gg and characterize whether ΓH, Gg is a tree, star graph, lollipop or a complete graph together with some properties of ΓH, Gg involving isomorphism of graphs. We also present certain relations between the number of edges in ΓH, Gg and certain generalized commuting probabilities of G which give some computing formulae for the number of edges in ΓH, Gg. Finally, we conclude this paper by deriving some bounds for the number of edges in ΓH, Gg.