Spectral aspects of commuting conjugacy class graph of finite groups

The commuting conjugacy class graph of a non-abelian group G, denoted by CCC(G), is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of G and two distinct vertices xG and yG are adjacent if there exists some elements x′∈xG and y′∈yG such that x′y′=y′x′. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups D2n,Q4m,U(n,m),V8n and SD8n. Our computation shows that CCC(G) is super integral for these groups. We compare various energies and as a consequence it is observed that CCC(G) satisfy E-LE Conjecture of Gutman et al. We also provide negative answer to a question posed by Dutta et al. comparing Laplacian and Signless Laplacian energy. Finally, we conclude this paper by characterizing the above mentioned groups G such that CCC(G) is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.