Super commuting graphs of finite groups and their Zagreb indices

TL;DR

This work introduces and analyzes $B$-super commuting graphs on finite groups, focusing on the cases where $B$ is equality, conjugacy, or same order. It provides explicit graph realizations for several nonabelian families by showing these graphs decompose as joins of complete graphs (the framework of generalized composition), enabling compact descriptions and easier computation of topological indices. The authors derive complete Zagreb indices $M_1$ and $M_2$ for the realized graphs via a general formula and demonstrate that the Hansen–Vukičević conjecture holds for all studied instances. Overall, the paper advances the interplay between group structure and graph invariants, offering compact graph models and confirming conjectural index relations across a broad class of groups.

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{č}evi{ć} conjecture.

Theorems & Definitions (43)

• Conjecture 1.1
• Proposition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof