Flow Polynomials and their Asymptotic Limits for Lattice Strip Graphs

Abstract

We present exact calculations of flow polynomials $F(G,q)$ for lattice strips of various fixed widths $L_y$ and arbitrarily great lengths $L_x$, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face $fl$ in the infinite-length limit. We study the zeros of $F(G,q)$ in the complex $q$ plane and determine exactly the asymptotic accumulation sets of these zeros ${\cal B}$ in the infinite-length limit for the various families of strips. The function $fl$ is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case $L_y=1$, the maximal point, $q_{cf}$, where ${\cal B}$ crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value $q_{cf}=3$ for the infinite square lattice.