Ground State Entropy of Potts Antiferromagnets on Homeomorphic Families of Strip Graphs

We present exact calculations of the zero-temperature partition function, and ground-state degeneracy (per site), $W$, for the $q$-state Potts antiferromagnet on a variety of homeomorphic families of planar strip graphs $G = (Ch)_{k_1,k_2,\Sigma,k,m}$,where$k_1$,$k_2$,$\Sigma$,and$k$ describe the homeomorphic structure, and $m$ denotes the length of the strip. Several different ways of taking the total number of vertices to infinity, by sending (i) $m\to \mathrm{\infty }$ with $k_1$, $k_2$, and $k$ fixed; (ii) $k_1$ and/or $k_2 \to \infty$with$m$,and$k$fixed;and(iii)$k \to \infty$with$m$and$p=k_1+k_2$fixedarestudiedandtherespectivelociofpoints${\cal B}$where$W$isnonanalyticinthecomplex$q$planearedetermined\mathrm{.}The${\cal B}${}^{\mathrm{\prime }}sforlimit(i)arecomprisedofarcswhichdonotencloseregionsinthe$q$planeand,formanyvaluesof$p$and$k$,includesupportfor$Re(q) < 0$. The $\mathcal{B}$ for limits (ii) and (iii) is the unit circle $\mathrm{\mid }q-1\mathrm{\mid }=1$.