Transforms Of Graphs,Resistance Distances And Spanning Trees

Let G=(V(G),E(G);?)be a connected edge-weighted graph with an edge-weighted function?:E(G)?R⁺.If we regarded G as an electrical network and the weight of each edge is the conductance,then the resistance distance between any two vertices of G is defined as the effective resistance between them in this electrical network.If G=(V(G),E(G);?)is a vertex-weighted connected graph with the vertex-weighted function?:V(G)?R⁺.Then it results in an edge-weighted graph and the weight of each edge e=uv equals?(e)=?(u)?(v).In this paper,by using the methods of electric networks such as the principle of substitution,(generalized)Delta-Wye transformation and series-parallel rules,et al.,and the sum rules of the effective resistance,firstly,we obtain the explicit formulae for the resistances of a vertex-weighted complete multipartite graph and give a new proof of a formula on the weighted enumeration of spanning trees of a vertex-weighted complete multipartite graph.Then,we obtain the K_m,n-double star transform similar to the gener-alized Delta-Wye transform.Using this transform,we obtain the formula for the number of spanning trees of a generalized join graph and count spanning trees of some special generalized join graphs.Finally,we consider the resistance distances of line and middle graphs of a general graph G,and prove that the resistance distances of line and middle graphs are determined by the resistances of a weighted subdivision graph of G.Particu-larly,we obtain formulae for the resistance distances of line and middle graphs when G is regular or semi-regular.