Super commuting graphs of finite groups and their Zagreb indices

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 $g^{\prime }\in [g]$ and $h^{\prime }\in [h]$ such that $g^{\prime }$ commutes with $h^{\prime }$, 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{č}evi{ć} conjecture.

22 pages