Degree-based topological indices of order super commuting graphs of finite groups

Abstract

Consider $\mathrm{D}$ to be an equivalence relation on a group $\mathscr{H}$ and let $\Psi$ be a graph type.
Given the equivalence relation $\mathrm{D}$, let $[y]$ denote the equivalence class of an element $y$.
The vertex set $V(\Psi)$ of the $\mathrm{D}$-super $\Psi$ graph of $\mathscr{H}$ represents an undirected graph such that two distinct vertices $y, z \in \Psi$ are adjacent if $[y] = [z]$, or if there exist elements $y^{\prime} \in [y]$ and $z^{\prime} \in [z]$ such that $y^{\prime}$ and $z^{\prime}$ are adjacent in the $\Psi$ graph of $\mathscr{H}$. In this study, we compute the degree-based topological indices of order super commuting graphs corresponding to various finite non-abelian groups, including dihedral, generalized quaternion, semidihedral, quasidihedral, $V_{8\vartheta}$, and $U_{6\vartheta}$ groups.
This work lies in the systematic derivation of closed-form expressions for these indices over new classes of order super commuting graphs.
The obtained results provide a deeper understanding of the structural properties and interrelations among these finite non-abelian groups through their topological descriptors.