Relative g-noncommuting graph of finite groups

TL;DR

This paper introduces the relative g-noncommuting graph of finite groups, providing formulas for vertex degrees, characterizations of graph types, and relations to generalized commuting probabilities, with bounds on edges.

Contribution

It defines and analyzes the structure of the relative g-noncommuting graph, including formulas, characterizations, and connections to group properties, which is a novel approach.

Findings

Formulas for vertex degrees in the graph

Characterizations of when the graph is a tree, star, lollipop, or complete

Relations between edges and generalized commuting probabilities

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 $\Gamma_{H, G}^g$.