Signless Laplacian energies of non-commuting graphs of finite groups and related results

The non-commuting graph of a non-abelian group G with center Z(G) is a simple undirected graph whose vertex set is G\Z(G) and two vertices x,y are adjacent if xy not equal yx. In this paper, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of certain finite non-abelian groups. We obtain several conditions such that the non-commuting graph of G is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the hyperenergetic and hypoenergetic properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic.