Eigenvalues of transition weight matrix for a family of weighted networks

In this article, we design a family of scale-free networks and study its random target access time and weighted spanning trees through the eigenvalues of transition weight matrix. First, we build a type of fractal network with a weight factor r r and a parameter m m . Then, we obtain all the eigenvalues of its transition weight matrix by revealing the recursive relationship between eigenvalues in every two consecutive time steps and obtain the multiplicities corresponding to these eigenvalues. Furthermore, we provide a closed-form expression of the random target access time for the network studied. The obtained results show that the random target access is not affected by the weight; it is only affected by parameters m m and t t . Finally, we also enumerate the weighted spanning trees of the studied networks through the obtained eigenvalues.