6. Laplace and signless laplace spectrum of commute graph of finite group
The commute graphs of a finite non-abelian groups G with center Z(G), indicated by ΓG, is the basic non-directed graphs whose vertices sets are G Z(G), and two discrete vertex x and y are together if and only if xy = yx. The finite non-abelian groups G are known as superficial integral if the spectrums, Laplace spectrums and signless Laplace spectrums of their commute graphs contains only integer. In this research work, authors initially calculate Laplace spectrums and sign-less Laplace spectrums of many families of finitely non-abelian group and achieve that this group is superior integrals. As an implementation of the obtained results, the authors obtained few positive ration number r so that G is superior integrals if commutative degrees of G is r. Finally, it is shown that G is superior integrals if G is not the iso-morphic to S4 and their commute graphs are in plane. The authors concluded the research paper indicating that G is superior integrals if their commute graphs are toroidal.