The Number Of Spanning Trees On Post-crtically Finite Self-similar Lattices

In this paper,we study the problem of counting the weighted spanning trees of post-critically finite(p.c.f.)self-similar lattices equipped with a harmonic structure.Let X0 be a complete graph with ? distinguished vertices,and let the sequence of p.c.f.self-similar lattices {Xn}n?0 be constructed by an iterated function system(i.f.s.)of contractive similarities {Fi,i=1,2,…,N}satisfying(?),n?1.We endow the sequence aforementioned with a harmonic structure(D,r),where-D is a Laplace matrix which makes all edges of the complete graph X0 be equipped with corresponding conductances denoted as vector c0,and each component of the vector r=(r1,r2,…,rN)is called a resistance scaling factor.For different i and j,ri and rj can be different.The main tool that we will apply is the technique developed by E.Teufl and S.Wagner which says that,if a part of a graph network is replaced by an electrically equivalent one,then the number of(weighted)spanning trees of the new graph network changes by a factor depending on the substituted graphs comparing to the number of spanning trees of the original graph network.Using this,we obtain the recurrence relation of the numbers of the(weighted)spanning trees,which determines an exact,computational formula of weighted spanning trees of{Xn}n?0.Noticing that there is a natural relationship between the eigenvalues of the Laplace matrix and the number of weighted spanning trees on an electrical network,this provides a new idea and perspective for the study of Laplace operator on p.c.f.fractals.