Properties and applications of Laplacian spectra for Koch networks

For a network, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics. In this paper, we study the Laplacian spectra of a family of Koch networks with scale-free and small-world properties. We derive some recursive relations between the Laplacian characteristic polynomials of Koch networks and their subgraphs at different iterations. 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, global mean first-passage time and average path length of the family of Koch networks. Finally, we determine the number of spanning forests under certain conditions. We expect that our method can be adapted to other types of self-similar networks.