Chromatic polynomials and toroidal graphs

The chromatic polynomials of some families of quadrangulations of the torus can be found explicitly. The method, known as ‘bracelet theory’ is based on a decomposition in terms of representations of the symmetric group. The results are particularly appropriate for studying the limit curves of the chromatic roots of these families. In this paper these techniques are applied to a family of quadrangulations with chromatic number 3, and a simple parametric equation for the limit curve is obtained. The results are in complete agreement with experimental evidence

Algebraic number-theoretic properties of graph and matroid polynomials

PhDThis thesis is an investigation into the algebraic number-theoretical properties of certain polynomial invariants of graphs and matroids. The bulk of the work concerns chromatic polynomials of graphs, and was motivated by two conjectures proposed during a 2008 Newton Institute workshop on combinatorics and statistical mechanics. The first of these predicts that, given any algebraic integer, there is some natural number such that the sum of the two is the zero of a chromatic polynomial (chromatic root); the second that every positive integer multiple of a chromatic root is also a chromatic root. We compute general formulae for the chromatic polynomials of two large families of graphs, and use these to provide partial proofs of each of these conjectures. We also investigate certain correspondences between the abstract structure of graphs and the splitting fields of their chromatic polynomials. The final chapter concerns the much more general multivariate Tutte polynomials—or Potts model partition functions—of matroids. We give three separate proofs that the Galois group of every such polynomial is a direct product of symmetric groups, and conjecture that an analogous result holds for the classical bivariate Tutte polynomial

Algebraic methods for chromatic polynomials

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

Chromatic invariants of signed graphs

AbstractWe continue the study initiated in “Signed graph coloring” of the chromatic and Whitney polynomials of signed graphs. In this article we prove and apply to examples three types of general theorem which have no analogs for ordinary graph coloring. First is a balanced expansion theorem which reduces calculation of the chromatic and Whitney polynomials to that of the simpler balanced polynomials. Second is a group of formulas based on counting colorings by their magnitudes or their signs; among them are a combinatorial interpretation of signed coloring (which implies an equivalence between proper colorings of certain signed graphs and matching in ordinary graphs) and a signed-graphic switching formula (which for instance gives the polynomials of a two-graph in terms of those of its associated ordinary graphs). Third are addition/deletion formulas obtained by constructing one signed graph from another through adding and removing arcs; one such formula expresses the chromatic polynomial as a combination of those of ordinary graphs, while another (in one example) yields a complementation formula for ordinary matchings. The examples treated are the sign-symmetric graphs (among them in effect the classical root systems), all-negative graphs (corresponding to the even-cycle graphic matroid), signed complete graphs (equivalent to two-graphs), and two varieties of signed graphs associated with matchings and colorings of ordinary graphs. Our results are interpreted as counting the acyclic orientations of a signed graph; geometrically this means counting the faces of the corresponding arrangement of hyperplanes or zonotope