Commuting Conjugacy Class Graphs of Finite Groups and the Hansen-Vuki\v{c}evi\'c Conjecture

Abstract

In this work, we compute the first and second Zagreb indices for the commuting conjugacy class graphs associated with finite groups. We identify multiple classes of finite groups whose commuting conjugacy class graphs are shown to satisfy the Hansen-Vuki{\v{c}}evi{\'c} conjecture. Specifically, we prove that the conjecture holds for the commuting conjugacy class graphs of dihedral groups ($D_{2m}$), dicyclic groups, semidihedral groups, and various other two-generator groups. Moreover, we examine the case where the quotient $G/Z(G)$ is isomorphic to $D_{2m}$, $\mathbb{Z}_p \times \mathbb{Z}_p$, a Frobenius group of order $pq$ or $p^2q$, or any group of order $p^3$, for primes $p$ and $q$. In each of these cases, we demonstrate that the corresponding commuting conjugacy class graph satisfies the Hansen-Vuki{\v{c}}evi{\'c} conjecture.