The number of spanning trees of a class of self-similar fractal models

Abstract

Computing the number of spanning trees of any general network model (graph) is almost too difficult to have no possibilities. As a vital constant of network model (graph), spanning trees number plays an important role not only on understanding some structural features, but also on determining some relevant dynamical properties, such as network security, random walks and percolation. Therefore, it becomes an interesting and attractive task taken more attention from various disciplines, including graph theory, theoretical computer science, physics and information science as well chemistry and so on. In this paper, our aim is to study the number of spanning trees of a class of self-similar fractal models. Firstly, we present the self-similar fractal model N(t) motivated from the well-known Koch curve. Next, due to its unique growth process and its spacial topological structure, we not only compute its average degree, which shows our model is sparse, but also capture a precise analytical solution for the total number of spanning trees of model N(t) by using both induction and iterative computational method. Finally, we obtain an approximate numerical value of its spanning tree entropy and then compare this value with ones of other network models researched previously. To conclude our work, we provide an opening and challenging problem that might be solved in the next coming days.