Inequalities involving energy and Laplacian energy of non-commuting graphs of finite groups

Let G be a finite non-abelian group and $${\Gamma }_{nc}(G)$$ Γ nc ( G ) be its non-commuting graph. In this paper, we compute spectrum and energy of $${\Gamma }_{nc}(G)$$ Γ nc ( G ) for certain classes of finite groups. As a consequence of our results we construct infinite families of integral complete r-partite graphs. We compare energy and Laplacian energy (denoted by $$E({\Gamma }_{nc}(G))$$ E ( Γ nc ( G ) ) and $$LE({\Gamma }_{nc}(G))$$ L E ( Γ nc ( G ) ) respectively) of $${\Gamma }_{nc}(G)$$ Γ nc ( G ) and conclude that $$E({\Gamma }_{nc}(G)) \le LE({\Gamma }_{nc}(G))$$ E ( Γ nc ( G ) ) ≤ L E ( Γ nc ( G ) ) for those groups except for some non-abelian groups of order pq. This shows that the conjecture posed in [Gutman, I., Abreu, N. M. M., Vinagre, C. T.M., Bonifacioa, A. S and Radenkovic, S. Relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem., 59: 343–354, (2008)] does not hold for non-commuting graphs of those finite groups, which also produces new families of counter examples to the above mentioned conjecture.