Reinfeld, Philipp Augustin (2003) Algebraic methods for chromatic polynomials. PhD thesis, The London School of Economics and Political Science (LSE).

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Abstract
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 blockdiagonal matrix. The multiplicities and the sizes of the blocks are obtained. Using a repeated inclusionexclusion argument the entries of the blocks can be calculated. In particular, from one of the inclusionexclusion 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 socalled generalized dodecahedra and four similar families of cubic graphs are obtained, and the limiting behaviour of their roots is discussed.
Item Type:  Thesis (PhD) 

Additional Information:  © 2003 Philipp Augustin Reinfeld 
Library of Congress subject classification:  Q Science > QA Mathematics 
Sets:  Departments > Mathematics Collections > LSE History of Thought theses 
URI:  http://etheses.lse.ac.uk/id/eprint/132 
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