Sombor Spectrum of Super Graphs defined on groups

TL;DR

This work investigates the $Sombor$ spectrum of $B$-super $A$ graphs defined on nonabelian groups, focusing on the dihedral, generalized quaternion, and semidihedral groups. By leveraging $\mathcal{R}$-compression, equitable partitions, and generalized-join representations, the authors derive characteristic polynomials and explicit eigenvalue structures for three base graphs (commuting, enhanced power, power) and their order/conjugacy variants. The main contributions are parity-sensitive spectral formulas and quotient-matrix descriptions that yield complete or near-complete spectra for a broad family of $B$-super $A$ graphs on $D_{2n}$, $Q_{4n}$, and $SD_{8n}$. These results enrich spectral graph theory on algebraic graphs and illuminate how group-theoretic structure shapes the Sombor spectrum, with potential implications for integrality properties and graph-invariant behavior. Overall, the paper provides a comprehensive, framework-driven catalog of $Sombor$ spectral data for a wide class of group-based super graphs.

Abstract

Given a simple graph $A$ on a group $G$ and an equivalence relation $B$ on $G$, the $B$ super $A$ graph is defined as a simple graph, whose vertex set is $G$ and two vertices $g$, $h$ are adjacent if either they are in the same equivalence class or there exist $g^{\prime} \in[g]$ and $h^{\prime} \in[h]$ such that $g^{\prime}$ and $h^{\prime}$ are adjacent in $A$. In the literature, the $B$ super $A$ graphs have been investigated by considering $A$ to be either power graph, enhanced power graph, or commuting graph and $B$ to be an equality, order or conjugacy relation. In this paper, we investigate the Sombor spectrums of these $B$ super $A$ graphs for certain non-abelian groups, viz. the dihedral group, generalized quaternion group and the semidihedral group, respectively.

Figures (9)

• Figure 1: Power graph $\mathcal{P}(Q_{4n})$
• Figure 2: Power graph $\mathcal{P}(SD_{8n})$
• Figure 3: Power graph $\mathcal{P}^o(Q_{4n})$, where $n$ is even
• Figure 4: Power graph $\mathcal{P}^o(Q_{4n})$, where $n$ is odd
• Figure 5: Power graph $\mathcal{P}^o(SD_{8n})$

Theorems & Definitions (58)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• proof
• Corollary 3.5
• proof