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All non-isomorphic graphs on 3 vertices and their chromatic polynomials, clockwise from the top. The independent 3-set: $k 3$ . An edge and a single vertex: $k 2 (k - 1)$ . The 3-path: $k (k - 1) 2$ . The 3-clique: $k (k - 1)(k - 2)$ .

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial by Hassler Whitney and W. T. Tutte, linking it to the Potts model of statistical physics.