$g$-noncommuting graph of a finite group relative to its subgroups

Abstract

Let $H$ be a subgroup of a finite non-abelian group $G$ and $g\in G$. Let $Z(H,G)=\{x\in H:xy=yx,\forall y\in G\}$. We introduce the graph ${\Delta }_{H,G}^g$ whose vertex set is $G\setminus Z(H,G)$ and two distinct vertices $x$ and $y$ are adjacent if $x\in H$ or $y\in H$ and $[x,y]\ne g,g^{-1}$, where $[x,y]=x^{-1}{y}^{-1}xy$. In this paper, we determine whether ${\Delta }_{H,G}^g$ is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

17 pages