Spectrum of super commuting graphs of some finite groups

Abstract

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