Laplacian Spectra of Power Graphs of Certain Finite Groups

Abstract

In this article, various aspects of Laplacian spectra of power graphs of finite cyclic, dicyclic and finite p-groups are considered. The algebraic connectivity is studied and the multiplicity of the Laplacian spectral radius is determined completely for power graphs of all of these groups. Then the equality of the vertex connectivity and the algebraic connectivity is characterized for power graphs of all of the above groups. Orders of dicyclic groups, for which their power graphs are Laplacian integral, are determined. Moreover, it is proved that the notion of equality of the vertex connectivity and the algebraic connectivity and the notion of Laplacian integral are equivalent for power graphs of dicyclic groups. All possible values of Laplacian eigenvalues are obtained for power graphs of finite p-groups. This shows that power graphs of finite p-groups are Laplacian integral.