Super commuting graphs of finite groups and their Zagreb indices

TL;DR

This paper introduces super commuting graphs based on various equivalence relations in finite groups, analyzes their structure for specific groups, and computes their Zagreb indices, confirming a conjecture.

Contribution

It defines and studies super commuting graphs for different equivalence relations in finite groups and computes their Zagreb indices, linking graph theory with group properties.

Findings

Super commuting graphs' structures are characterized for several non-abelian groups.

Zagreb indices of these graphs are computed explicitly.

The graphs satisfy Hansen-Vuki{ extcrv}evi{\'c} conjecture.

Abstract

Let $B$ be an equivalence relation defined on a finite group $G$. The $B$ super commuting graph on $G$ is a graph whose vertex set is $G$ and two distinct vertices $g$ and $h$ are adjacent if either $[g] = [h]$ or there exist $g' \in [g]$ and $h' \in [h]$ such that $g'$ commutes with $h'$, where $[g]$ is the $B$-equivalence class of $g \in G$. Considering $B$ as the equality, conjugacy and same order relations on $G$, in this article, we discuss the graph structures of equality/conjugacy/order super commuting graphs of certain well-known families of non-abelian groups viz. dihedral groups, dicyclic groups, semidihedral groups, quasidihedral groups, the groups $U_{6n}, V_{8n}, M_{2mn}$ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vuki{\v{c}}evi{\'c} conjecture.