Spectrum of conjugacy and order super commuting graphs of some finite groups

Abstract

Let $\Gamma$ be a simple finite graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. Let $\mathcal{R}$ be an equivalence relation on $V(\Gamma)$. The $\mathcal{R}$-super $\Gamma$ graph $\Gamma^{\mathcal{R}}$ is a simple graph with vertex set $V(\Gamma)$ and two distinct vertices are adjacent if either they are in the same $\mathcal{R}$-equivalence class or there are elements in their respective $\mathcal{R}$-equivalence classes that are adjacent in the original graph $\Gamma$. We first show that $\Gamma^{\mathcal{R}}$ is a generalized join of some complete graphs and using this we obtain the adjacency and Laplacian spectrum of conjugacy and order super commuting graphs of dihedral group $D_{2n}\; (n\geq 3)$, generalized quaternion group $Q_{4m} \;(m\geq 2)$ and the nonabelian group $\mathbb Z_p \rtimes \mathbb Z_q$ of order $pq$, where $p$ and $q$ are distinct primes with $q|p-1$.