Zagreb indices of commuting and non-commuting graphs of finite groups and Hansen-Vuki\v{c}evi\'c conjecture

TL;DR

This paper calculates Zagreb indices for commuting and non-commuting graphs of finite groups and identifies classes of groups that satisfy the Hansen-Vukićević conjecture, advancing understanding in algebraic graph theory.

Contribution

It introduces explicit calculations of Zagreb indices for these graphs and characterizes finite groups whose graphs meet the Hansen-Vukićević conjecture.

Findings

Computed Zagreb indices for various finite groups' graphs.

Identified classes of groups satisfying the Hansen-Vukićević conjecture.

Provided new insights into the structure of finite groups via graph invariants.

Abstract

In this paper we compute first and second Zagreb indices of commuting and non-commuting graphs of finite groups and determine several classes of finite groups such that their commuting and non-commuting graphs satisfy Hansen-Vuki\v{c}evi\'c conjecture.