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
