On distance spectra, energies and Wiener index of non-commuting conjugacy class graphs

Abstract

The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of a finite non-abelian group $H$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $H$ and two vertices, $a^H$ and $b^H$ are adjacent if $a'b' \ne b'a'$ for all $a' \in a^H$ and $b' \in b^H$. In this paper, we compute distance spectrum, distance Laplacian spectrum, distance signless Laplacian spectrum along with their respective energies and Wiener index of NCCC-graphs of $H$ when the central quotient of $H$ is isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ (for any prime $p$) or $D_{2n}$ (for any integer $n \geq 3$). As a consequence, we compute various distance spectra, energies and Wiener index of NCCC-graphs of the dihedral group, dicyclic group, semidihedral group along with the groups $U_{(n,m)}$, $U_{6n}$ and $V_{8n}$. Thus we obtain sequences of positive integers that can be realized as Wiener index of NCCC-graphs of certain groups. In particular, we solve Inverse Wiener index Problem for NCCC-graphs of groups when $n$ is a perfect square. We further characterize the above-mentioned groups such that their NCCC-graphs are D-integral, DL-integral and DQ-integral. We also compare various distance energies of NCCC-graphs of the above mentioned groups and characterize those groups subject to the inequalities involving various distance energies.