Spectrum of super commuting graphs of some finite groups

Abstract Let Γ be a simple finite graph with vertex set V () V (Γ) and edge set E () E (Γ). Let R R be an equivalence relation on V () V (Γ). The R R -super Γ graph ^ {R} Γ R is a simple graph with vertex set V () 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. We first show that ^ {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 D₂₍\; (n 3) D 2 n (n ≥ 3), generalized quaternion group Q₄₌ \; (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).