  Let $G$ be a finite non-abelian group and ${\Gamma}_{nc}(G)$ be its
non-commuting graph. In this paper, we compute spectrum and energy of
${\Gamma}_{nc}(G)$ 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
$E({\Gamma}_{nc}(G))$ and $LE({\Gamma}_{nc}(G))$ respectively) of
${\Gamma}_{nc}(G)$ and conclude that $E({\Gamma}_{nc}(G)) \leq
LE({\Gamma}_{nc}(G))$ 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 certain finite groups, which
also produces new families of counter examples to the above mentioned
conjecture.
