Combinatorial Greenʼs function of a graph and applications to networks

Given a finite weighted graph G and its Laplacian matrix L, the combinatorial Greenʼs function G of G is defined to be the inverse of L+J, where J is the matrix each of whose entries is 1. We prove the following intriguing identities involving the entries in G=(gij) whose rows and columns are indexed by the vertices of G: gaa+gbb−gab−gba=κ(Ga⁎b)/κ(G), where κ(G) is the complexity or tree-number of G, and Ga⁎b is obtained from G by identifying two vertices a and b. As an application, we derive a simple combinatorial formula for the resistance between two arbitrary nodes in a finite resistor network. Applications to other similar networks are also discussed.