Chromatic polynomials for lattice strips with cyclic boundary conditions

The zero-temperature q-state Potts model partition function for a lattice s trip of fixed width L-y and arbitrary length L-x has the form P(G,q) = Sigm a (NG,lambda)(j=1) c(G), (j)(lambda (G,j))(Lx) and its equivalent chromatic polynomial for this graph. We present exact zero-temperature partition fun ctions for strips of several lattices with (FBCy,PBCx), i.e., cyclic, bound ary conditions. In particular, the chromatic polynomial of a family of gene ralized dodecahedra graphs is calculated. The coefficient c(G,j) of degree d in q is c((d)) = U-2d(rootq/2), where U-n(x) is the Chebyshev polynomial of the second kind. We also present the chromatic polynomial for the strip of the square lattice with (PBCy,PBCx), i.e., toroidal, boundary conditions and width L-y = 4 With the property that each set of four vertical vertice s forms a tetrahedron. A number of interesting and novel features of the co ntinuous accumulation set of the chromatic zeros, R are found. (C) 2001 Pub lished by Elsevier Science B.V.