Complex roots of chromatic polynomials
Seminar Room 1, Newton Institute
I shall begin by explaining how the theory of representations of the symmetric group can be applied to the transfer matrix. This leads to explicit formulae for the chromatic polynomials of families of graphs, in which the terms correspond to partitions of positive integers.
The formulae are well-suited to the application of the Beraha-Kahane-Weiss theorem, describing the limit points of zeros of the polynomials. In simple cases the individual terms can be written explicitly as powers of polynomials, and the resulting limit curves are (parts of) closed curves. In the general case the curves can have end-points and singularities, and I shall discuss some of the interesting phenomena that can occur.
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