Foster's Formulas via Probability and the Kirchhoff Index

Abstract

Building on a probabilistic proof of Foster’s first formula given by Tetali (1994), we prove an elementary identity for the expected hitting times of an ergodic N-state Markov chain which yields as a corollary Foster’s second formula for electrical networks, namely

$$\sum {R_{ij} } \frac{{C_{iv} C_{vj} }}{{C_v }} = N - 2,$$

where R _ij is the effective resistance, as computed by means of Ohm’s law, measured across the endpoints of two adjacent edges (i,v) and (i,v), C _iv and C _iv are the conductances of these edges, C _v is the sum of all conductances emanating from the common vertex v, and the sum on the left hand side of (1) is taken over all adjacent edges. We show how to extend Foster’s first and second formulas. As an application, we show how to use a “third formula” to compute the Kirchhoff index of a class of graphs with diameter 3.