The Number of Spanning Trees in some Special Self-Similar Graphs

Abstract: In this paper, we introduce two families of planar and self-similar graphs which have small-world properties. The constructed models are based on an iterative process where each step of a certain formulation of modules results in a final graph with a self-similar structure. The number of spanning trees of a graph is one of the most graph-theoretical parameters, where its applications range from the theory of networks to theoretical chemistry. Two explicit formulas are introduced for the number of spanning trees for the two models. With explicit formulas for some of their topological parameters as well.