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)k1,k2,Σ,k,m, where k1, k2, Σ, 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→∞ with k1, k2, and k fixed; (ii) k1 and/or k2→∞ with m, and k fixed; and (iii) k→∞ with m and p=k1+k2 fixed are studied and the respective loci of points where W is nonanalytic in the complex q plane are determined. The ’s for limit (i) are comprised of arcs which do not enclose regions in the q plane and, for many values of p and k, include support for Re(q)<0. The for limits (ii) and (iii) is the unit circle q−1=1.