Laplacian Spectrum and Energy of Non-Commuting Graphs of Finite Rings

We compute spectrum, energy, Laplacian spectrum/ energy and signless Laplacian spectrum/energy of non-commuting graphs of certain finite non-commutative rings. In particular, we consider finite rings R such that |R| = p², p³, p⁴, p⁵, p²q and p³q, where p and q are primes. Further, we consider n-centralizer finite\\ rings for $n \, = \,4, \, 5$ \, and \, $p \,+ \,2$; \, more generally, finite rings with central quotients isomorphic to Zₚ × Zₚ. Our computations reveal that non-commuting graphs of these rings are L-integral. We also determine whether non-commuting graphs of these rings are integral, Q-integral, hyperenergetic, L-hyperenergetic or Q-hyperenergetic.