Tutte Polynomials and Related Asymptotic Limiting Functions for Recursive Families of Graphs

Abstract

We prove several theorems concerning Tutte polynomials $T(G,x,y)$ for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the Potts model partition function of the $q$-state Potts model, $Z(G,q,v)$, where $v$ is a temperature-dependent variable. We determine the structure of the Tutte polynomial for a cyclic clan graph $G[(K_r)_m,L=jn]$ comprised of a chain of $m$ copies of the complete graph $K_r$ such that the linkage $L$ between each successive pair of $K_r$'s is a join $jn$, and $r$ and $m$ are arbitrary. The explicit calculation of the case $r=3$ (for arbitrary $m$) is presented. The continuous accumulation set of the zeros of $Z$ in the limit $m \to \infty$ is considered. Further, we present calculations of two special cases of Tutte polynomials, namely, flow and reliability polynomials, for cyclic clan graphs and discuss the respective continuous accumulation sets of their zeros in the limit $m \to \infty$. Special valuations of Tutte polynomials give enumerations of spanning trees and acyclic orientations. Two theorems are presented that determine the number of spanning trees on $G[(K_r)_m,jn]$ and $G[(K_r)_m,id]$, where $L=id$ means that the identity linkage. We report calculations of the number of acyclic orientations for strips of the square lattice and use these to obtain an improved lower bound on the exponential growth rate of the number of these acyclic orientations.