Evaluations of Tutte polynomials of regular graphs

Abstract

Let TG(x,y) be the Tutte polynomial of a graph G. In this paper we show that if (Gn)n is a sequence of d-regular graphs with girth g(Gn)→∞, then for x≥1 and 0≤y≤1 we havelimn→∞⁡TGn(x,y)1/v(Gn)=td(x,y), wheretd(x,y)={(d−1)((d−1)2(d−1)2−x)d/2−1ifx≤d−1and0≤y≤1,x(1+1x−1)d/2−1ifx>d−1and0≤y≤1. If (Gn)n is a sequence of random d-regular graphs, then the same statement holds true asymptotically almost surely.This theorem generalizes results of McKay (x=1,y=1, spanning trees of random d-regular graphs) and Lyons (x=1,y=1, spanning trees of large-girth d-regular graphs). Interesting special cases are TG(2,1) counting the number of spanning forests, and TG(2,0) counting the number of acyclic orientations.