An Electrical Lemma

Let s and t be distinct vertices of a connected graph G. The notation and definitions used here follow the excellent text of Bollabas [1]. Next let N be the number of spanning trees of G while F(s, t) = F is the number of spanning forests with two components, one containing s and one containing t. Call such a forest a thicket. The lemma in question states that Rs,t = F/N where Rt is the resistance of G between s and t if each edge of G represents a one ohm resistor. Since the right-hand side is a strictly graphical quantity this is a fascinating result which deserves to be better known. By way of illustration, look at the smallest example not immediately solved by the series and parallel laws.