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 $x \in H$ or $y \in H$ and $[x,y] \neq g, g^{-1}$. We denote this graph by $\Gamma_{H, G}^g$. In this paper, we obtain computing formulae for degree of any vertex in $\Gamma_{H, G}^g$ and characterize whether $\Gamma_{H, G}^g$ is a tree, star graph, lollipop or a complete graph together with some properties of $\Gamma_{H, G}^g$ involving isomorphism of graphs. We also present certain relations between the number of edges in $\Gamma_{H, G}^g$ and certain generalized commuting probabilities of $G$ which give some computing formulae for the number of edges in $\Gamma_{H, G}^g$. Finally, we conclude this paper by deriving some bounds for the number of edges in $\Gamma_{H, G}^g$.