Short proof of a theorem of Brylawski on the coefficients of the Tutte polynomial

Abstract

The Tutte polynomial T G (x, y) of a graph G = (V, E) with v(G) vertices and e(G) edges is defined aswhere k(A) denotes the number of connected components of the graph (V, A).By writing the Tutte polynomial as T G (x, y) = i,j t ij x i y j , the coefficients satisfy Brylawski's linear relations, namely that for any h < e(G), we have 2010 Mathematics Subject Classification. Primary: 05C31.