Spectrum of super commuting graphs of some finite groups

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 super commuting graphs and order super commuting graphs of dihedral group \(D_{2n}\; (n\ge 3)\), generalized quaternion group \(Q_{4m} \;(m\ge 2)\) and the nonabelian group \(\mathbb {Z}_p \rtimes \mathbb {Z}_q\) of order pq, where p and q are distinct primes with \(q|(p-1)\).