Relative g-noncommuting graph of finite groups

2020/08/10 by Sharma, Monalisha, Nath, Rajat Kanti
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2008.04123

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

Related

• g-noncommuting graph of a finite group relative to its subgroups
• Aspects of the commuting graph
• Graphs encoding the generating properties of a finite group
• Proof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups
• Generalized non-coprime graphs of groups