Study on Applications of Laplacian spectra for a network

Systems composing of dynamical units are ubiquitous in nature, ranging from physical to technological, and to biological field. These systems can be naturally described by networks, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics for a network. In this paper, we study the Laplacian spectra of a family with scale-free and small-world properties. Based on the obtained recurrence relations, we determine explicitly the product of all nonzero Laplacian eigenvalues, as well as the sum of the reciprocals of these eigenvalues. Then, using these results, we further evaluate the number of spanning trees, Kirchhoff index.