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

The non-commuting graph of a non-abelian group Formula: see text with center Formula: see text is a simple undirected graph whose vertex set is Formula: see text and two vertices Formula: see text are adjacent if Formula: see text. 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 Formula: see text 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.