Chromatic polynomials

Question

I understand that chromatic polynomials give the number of possible colorings for a graph with $t$ colors. For example, the chromatic polynomial for a complete graph with $n$ vertices is $t(t-1)\cdots(t-n+1)$ because there are $t$ ways to choose the first color, $t-1$ ways to choose the 2nd color, etc. and the same color cannot be used twice because all the vertices are connected.

My question is: how do you derive (in general) the chromatic polynomial for a graph? Specifically, what is the chromatic polynomial for the Peterson graph and how do you derive it?

P.S. Also I've seen "characteristic polynomial" substituted for "chromatic polynomial" sometimes. Are the two terms equivalent? If not, how are they different?

Many thanks :)

Answer 1

The chromatic polynomial of the Petersen graph is $t \left( t-1 \right) \left( t-2 \right) \left( {t}^{7}-12\,{t}^{6}+ 67\,{t}^{5}-230\,{t}^{4}+529\,{t}^{3}-814\,{t}^{2}+775\,t-352 \right) $. You can look it up in Wikipedia, or ask Maple for it. I don't think it is easy to compute by hand. There are recursive formulas for the chromatic polynomial using edge deletion and contraction.

The characteristic polynomial is not the same as the chromatic polynomial. It is the characteristic polynomial (in the sense of linear algebra) of the adjacency matrix of the graph.

Answer 2

The relation between chromatic polynomial and characteristic polynomial of a matroid is: Let $M$ be the graph matroid of a simple graph $G$. Then $\chi_G(q) = q^c\chi_M(q)$, where $c$ is the number of connected components of $G$, see for example the paper on page 2.