Abstract
Laplacian spectra and their applications are involved in diverse theoretical problems on complex networks. In this paper, we considered the properties of the Laplacian matrices for a family of scale-free small-world networks, controlled by two parameters m and r and called (m,r)-Koch networks. In particular, we derived the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Furthermore, these results were used to deal with various problems that often arise in the study of networks including Kirchhoff index, global mean-first passage time, average path length and the number of spanning trees.
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