Let a simple graph G have n vertices, chromatic polynomial P(x), and chromatic number χ. Then P(G) can be written as P(G) = sum_(i = 0)^h a_i·(x)_(p - i), where h = n - χ and (x)_k is a falling factorial, and the polynomial σ(G) = sum_(i = 0)^h a_i x^(h - i) is known as the σ-polynomial (Frucht and Giudici 1983; Li et al. 1987, p. 265). σ-polynomials for a number of simple graphs are summarized in the following table.

chromatic polynomial | Royle graphs