  In this article, various aspects of Laplacian spectra of power graphs of
finite cyclic, dicyclic and finite $p$-groups are studied. Algebraic
connectivity of power graphs of the above groups are considered and determined
completely for that of finite $p$-groups. Further, the multiplicity of
Laplacian spectral radius of power graphs of the above groups are studied and
determined completely for that of dicyclic and finite $p$-groups. 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.
