Random walks and the effective resistance sum rules

In this paper, using the intimate relations between random walks and electrical networks, we first prove the following effective resistance local sum rules: ciΩij+∑k∈Γ(i)cik(Ωik−Ωjk)=2, where ΩijΩij is the effective resistance between vertices ii and jj, cikcik is the conductance of the edge, Γ(i)Γ(i) is the neighbor set of ii, and ci=∑k∈Γ(i)cikci=∑k∈Γ(i)cik. Then we show that from the above rules we can deduce many other local sum rules, including the well-known Foster’s kk-th formula. Finally, using the above local sum rules, for several kinds of electrical networks, we give the explicit expressions for the effective resistance between two arbitrary vertices.