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

TL;DR

This paper introduces a new graph associated with a finite group and its subgroup, analyzing its structural properties such as being a tree, diameter, and connectivity, with specific focus on dihedral groups.

Contribution

It defines the $g$-noncommuting graph for finite groups relative to subgroups and investigates its properties, including criteria for being a tree and its diameter and connectivity.

Findings

Determines conditions under which the graph is a tree.

Analyzes the diameter and connectivity of the graph.

Provides specific results for dihedral groups.

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 $[x,y] \neq g, g^{-1}$, 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.