My research interests lie primarily in the connections between different areas of mathematics and between mathematics and the outside world. I am a combinatorialist, fascinated by the discrete, with graph theory being a first love.

Combinatorial Polynomials

Polynomials arise from graphs in a variety of different ways, including:

  • The chromatic polynomial enumerates how many ways you can properly colour the vertices of a graph with k colours.
  • The all-terminal reliability of a graph reflects the probability that all vertices can comminucate, given that vertices are always operational, but each edge is independently operational with probability p.
  • The independence polynomial of a graph is the generating function of the number of independent sets of each size.

My particular interest lies in the nature and location of the roots of such polynomials. The distribution of the roots in the complex plane can inform you of combinatorial propertiues of the underlying coefficients, including unimodality. Often there are underlying simplicial complexes for which the polynomials are simple evaluations of f-polynomials (or face polynomials) of the complex, and this puts many of these polynomials in a common framework. The images generated by the roots of various graph polynomials can be quite beautiful, and sometimes fractal in nature.

Also, when the underlying simplicial complexes have a particular property known as shellable, then there is a deep connection to commutative algebra. The study of the associated ideals are not only interesting in their own right, but also point to interesting questions involving Grober bases over finite fields.

Other polynomials of interest are

  • the strongly connected reliability polynomials (the natural extension of all-terminal reliability to digraphs),
  • neighbourhood polynomials, which are generating functions for the subsets of vertices with a common neighbour.,
  • ideal polynomials, which arise as the generating funciton for the number of ideals (downward closed sets) of each cardinality in a partial order,
  • generating polynomials for the number of g-convex sets of a graph, and
  • sparse, lacunary binomial-type polynomials

Well Covered Graphs and Vector Spaces

A graph is well covered if every maximal independent set has the same size. There is much known about the structure of such graphs. In conjuction with Richard Hoshino I looked at well covered circulants (those are graphs on the integers mod n where adjacency corresponds to the difference being in some set S, not containing 0, with -S = S).

One can extend the notion of well coveredness algebfraically as follows. Suppose G is a graph and k a field. A weighting is a function from the vertex set of G into k; such a weighting is a well covered weighting if the sum of the weights over any maximal independent set is constant. The weightings form a vector space over k, and questions of dimension and bases are intriguing.

Mathematics and Music

There are a number of ways that mathematics intersects with musicand I have applied mathematics to a number of areas of music:

  • Fourier transforms were used to uncover how the Beatles played the opening chord of A Hard Day's Night, and prove that George Harrison's solo in the same song was recorded at half-speed, down the octave.
  • Graph Colourings help explain why the blues chord progression is so universally appreciated.
  • The perception of pitch and rhythm along logarithmic scales explain why George Martin couldn't have improved his famous edit of the Beatles' Strawberry Fields Forever (joint work with Robert Dawson of St. Marys University).