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

Abstract

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.