The Clebsch graph (of degree 5) is 4-colourable, so the roots of its chromatic polynomial are 0, 1, 2, 3, and a dozen other non-trivial algebraic numbers. With most graphs, these roots are scattered around the complex plane more-or-less at random (close to the origin, the real part tends to be positive), but those of the Clebsch graph very obviously lie on a curve.
Assume the roots $(x_i, y_i)$ lie on a curve described by $x=f(y)$. Obviously $f$ is an even function. Plugging the data points into Excel, a degree 4 polynomial approximates it pretty well ($R^2>0.99$), but with only six pairs of data points it's hard to be sure. It could be something else, like a hyperbolic cosine. How do you go about checking something like this? And more generally, are there other examples of the roots of a high-degree polynomial having a pattern like this?
I've looked at a few other graphs. The Paley graph on 13 points also has non-trivial chromatic roots like this - this time four pairs of complex roots. These graphs share two properties: they're strongly regular, and $m$ copies of them cover the edges of $K_n$ (where $n$ is the order of the graph). The Paley graph is self-complementary, so obviously $m=2$; as for Clebsch, $m=3$ (which shows up in the proof that the Ramsey number $R(3,3,3)=17$). The roots of Paley(9) don't behave like this, though. Paley(5) is too small to get anything meaningful. (It's a pentagon; the chromatic roots of any cycle are on a circle centred at (1,0).
I'm using free software to get these polynomials, so that's about as big as I've got; Paley(17) would be the next graph I'd look at if I could, and more Paley graphs beyond that (maybe just for prime degrees).