A Randomised Approximation Algorithm for Counting the Number of Forests in Dense Graphs

r(A) = \V\-k(A), (2) where k(A) is the number of connected components of the graph with vertex set V and edge set A (including isolated vertices). It can immediately be seen from the above that the number of forests of a graph is equal to the value of the Tutte polynomial at the point (2,1). The Tutte polynomial contains many other invariants of fundamental importance in fields as diverse as statistical physics, knot theory and graph colourings. For example, the partition function of the q-state Potts model of statistical physics can be calculated from an evaluation of the Tutte polynomial along the hyperbola