Polychromatic polynomials Original Research Article

We refine the polychromate defined by Brylawski (J. Combin. Theory Ser. B 30 (1981) 233–246) to a graph invariant that takes account of a root vertex. Using the multiplicative structure that results, we show that this rooted polychromate distinguishes all non-isomorphic rooted trees. We also define an invariant finer than the polychromate — the strong polychromate — and conjecture that this invariant uniquely identifies almost all graphs.