Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2003-12

July 2003

Algebraic methods for chromatic polynomials

Philipp Reinfeld


The chromatic polynomials of certain families of graphs can be calculated by a transfer matrix method. The transfer matrix commutes with an action of the symmetric group on the colours. Using representation theory, it is shown that the matrix is equivalent to a block-diagonal matrix. The multiplicities and the sizes of the blocks are obtained.
Using a repeated inclusion-exclusion argument the entries of the blocks can be calculated. In particular, from one of the inclusion-exclusion arguments it follows that the transfer matrix can be written as a linear combination of operators which, in certain cases, form an algebra. The eigenvalues of the blocks can be inferred from this structure.
The form of the chromatic polynomials permits the use of a theorem by Beraha, Kahane and Weiss to determine the limiting behaviour of the roots. The theorem says that, apart from some isolated points, the roots approach certain curves in the complex plane. Some improvements have been made in the methods of calculating these curves.
Many examples are discussed in detail. In particular the chromatic polynomials of the family of the so-called generalized dodecahedra and four similar families of cubic graphs are obtained, and the limiting behaviour of their roots is discussed.

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