A Matrix Method for Chromatic Polynomials -ii 1. Introduction

The subject of this paper is the calculation of the chromatic polynomials of families of graphs that have cyclic symmetry. Generally speaking, there are no elegant methods in this eld. The standard method of deletion-and-contraction is both ineecient and inelegant: it requires exponentially many steps, and any symmetry that the graph possesses is destroyed at the very rst step. In this paper we consider families of graphs fG n g, where G n has n-fold cyclic symmetry, and we shall develop a method, introduced in 2], that uses this symmetry in a crucial way. There are two basic steps. First, we introduce a `compatibility operator' T, and apply the sieve principle to obtain an explicit formula for its action. Then we show that, in certain circumstances , it is possible to use this formula to construct invariant subspaces for T. Furthermore, the spectral decomposition of the action of T on these subspaces can be calculated (in principle). The resulting eigenvalues i and their multiplicities m i correspond to terms in the chromatic polynomial of G n , which takes the form It is signiicant that the eigenvalues occur inìevels', in such a way that the top levels are the simplest to compute and also the most important. It has been known for many years that results of this kind provide insight into physical processes. In this context, the behaviour of the zeros of P(G n ; z) as n ! 1 is of paramount importance. A result of Beraha, Kahane and Weiss 1] implies that the zeros approach curves deened by equations of the form j i (z)j = j j (z)j (i 6 = j): Shrock and Tsai 8] point out that these curves divide the complex plane into regions, in each of which the thermodynamic limit lim n!1 fP(G n ; z)g 1=n is a well-deened analytic function. A series of papers by Shrock and his colleagues (see 9] and the references given there) discusses the relationship between these ideas and the critical behaviour ofàntiferromagnets'. It appears that the curves that determine the critical regions are those corresponding to the eigenvalues at the top levels. The key elements of the method are described in Sections 2, 3, and 4. In Section 3 we obtain a general sieve formula for the action of T, and in Section 1