Spectrum of super commuting graphs of some finite groups

Let Γ Γ be a simple finite graph with vertex set V(Γ ) V ( Γ ) and edge set E(Γ ) E ( Γ ) . Let R R be an equivalence relation on V(Γ ) V ( Γ ) . The R R -super Γ Γ graph Γ R Γ R is a simple graph with vertex set V(Γ ) V ( Γ ) and two distinct vertices are adjacent if either they are in the same R R -equivalence class or there are elements in their respective R R -equivalence classes that are adjacent in the original graph Γ Γ . We first show that Γ R Γ 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 D₂ₙ\; (n≥ 3) D 2 n ( n ≥ 3 ) , generalized quaternion group Q₄ₘ \;(m≥ 2) Q 4 m ( m ≥ 2 ) and the nonabelian group Zₚ Zq Z p ⋊ Z q of order pq , where p and q are distinct primes with $$q|(p-1)$$ q | ( p - 1 ) .