Chromatic Polynomials and Representations of the Symmetric Group

The chromatic polynomial P (G; k) is the function which gives the number of ways of colouring a graph G when k colours are available. The fact that it is a polynomial function of k is essentially a consequence of the fact that, when k exceeds the number of vertices of G, not all the colours can be used. Another quite trivial property of the construction is that the names of the k colours are immaterial; in other words, if we are given a colouring, then any permutation of the colours produces another colouring. In this talk I shall outline some theoretical developments, based on these simple facts and some experimental observations about the complex roots of chromatic polynomials of ‘bracelets’. A ‘bracelet’ Gn = Gn(B, L) is formed by taking n copies of a graph B and joining each copy to the next by a set of links L (with n + 1 = 1 by convention). The chromatic polynomial of Gn can be expressed in the form