The chromatic invariant θ(G) of a connected graph G is the number of spanning trees of G that have internal activity 1 and external activity 0. For graphs other than the singleton graph (for which θ(K_1) = 1), it is also given by θ(G) = (-1)^n auto left match (d π_G)/(d x) right bracketing bar _(x = 1), where n = V(G) = left bracketing bar G right bracketing bar is the vertex count and π_G(x) is the chromatic polynomial of G. A connected graph G is separable iff θ(G) = 0 and is a series-parallel graph iff θ(G)<=1.

chromatic number | chromatic polynomial