Applications of Markov spectra for the weighted partition network by the substitution rule

The weighted self-similar network is introduced in an iterative way. In order to understand the topological properties of the self-similar network, we have done a lot of research in this field. Firstly, according to the symmetry feature of the self-similar network, we deduce the recursive relationship of its eigenvalues at two successive generations of the transition-weighted matrix. Then, we obtain eigenvalues of the Laplacian matrix from these two successive generations. Finally, we calculate an accurate expression for the eigentime identity and Kirchhoff index from the spectrum of the Laplacian matrix.