Eigentime identities for random walks on a family of treelike networks and polymer networks

In this paper, we investigate the eigentime identities quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on a family of treelike networks and the polymer networks. Firstly, for a family of treelike networks, it is shown that all their eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. We obtain the scalings of the eigentime identity on a family of treelike with network size Nn is NnlnNn. Then, for the polymer networks, we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities. Using the relationship between the generation and the next generation of eigenvalues we obtain the scalings of the eigentime identity on the polymer networks with network size Nn is NnlnNn. By comparing the eigentime identities on these two kinds of networks, their scalings with network size Nn are all NnlnNn.