Orbital Chromatic and Flow Roots

The chromatic polynomial PΓ(x) of a graph Γ is a polynomial whose value at the positive integer k is the number of proper kcolourings of Γ. If G is a group of automorphisms of Γ, then there is a polynomial OPΓ,G(x), whose value at the positive integer k is the number of orbits of G on proper k-colourings of Γ. It is known there are no real negative chromatic roots, but they are dense in [ 27 ,∞). Here we discuss the location of real orbital chromatic roots. We show, for example, that they are dense in R, but under certain hypotheses, there are zero-free regions. Our hypotheses include parity conditions on the elements of G and also some special types of graphs and groups. We also look at orbital flow roots. Here things are more complicated because the orbit count is given by a multivariate polynomial; but it has a natural univariate specialisation, and we show that the roots of these polynomials are dense in the negative real axis.