Spanning Trees And Effective Resistances Of 2-separable Graphs

Let G=V(G),E(G)be a connected graph with m edges ei=(ai,bi)where i=1,2,…,m.Suppose Ha1 b1,Ha2 b2,...,Hambm are m graphs,where ai and bi are two fixed vertices in Hai bu.Now we replace all of edges ei of G with graphs Hai bi for i=1,2,...,m.Then we get a new graph,which is called to be a 2-separable graph.There are so many large-scale networks in the context of statistical physics which are obtained by iteration of 2-separable graph operation.Gong and Li considered the enumerative problem of the spanning tree of a special 2-separable graph(i.e.,Ha1b1=Ha2b2=...=Hambm=Hab)and obtained the enumerative formula of spanning trees.In this paper,we consider firstly the enumerative problem of spanning trees of the general 2-separable graphs.As applications,we compute the numbers of spanning trees of some lattices in the context of statistical physics and biology.We regard a 2-separable graph as a 2-separable electrical network and discuss the computation of resistances between any two vertices in a 2-separable electrical networks.As applications,we compute the resistances of two types of edge corona graphs which are 2-separable.