Some new evaluations of the Tutte polynomial

Interpretations for evaluations of the Tutte polynomial T ( G ; x , y ) of a graph G are given at a number of points on the hyperbolae H q = { ( x , y ) | ( x - 1 ) ( y - 1 ) = q } , for q a positive integer—points at which there are usually no other similarly meaningful graphical interpretations. Further, when q is a prime power, an alternative interpretation for the evaluation of the Tutte polynomial at ( 1 - q , 0 ) is presented, more familiarly known as the point which gives the number of proper vertex q-colourings of G.