strong induction assume the statement is true for . Let G have edges. By the contraction-deletion principle, , Let , and . Hence .Since is obtained from G by removal of just one edge e, , so and thus the statement is true for k. The coefficient of is times the number of acyclic orientations that have a unique sink, at a specified, arbitrarily chosen vertex. The absolute values of coefficients of every chromatic polynomial form a log-concave sequence. The last property is generalized by the fact that if G is a k-clique-sum of and (i.e., a graph obtained by gluing the two at a clique on k vertices), then A graph G with n vertices is a tree if and only if Chromatic equivalence Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. Isomorphic graphs have the same chromatic polynomial, but non-isomorphic graphs can be chromatically equivalent. For example, all trees on n vertices have the same chromatic polynomial. In particular, is the chromatic polynomial of both the claw graph and the path graph on 4 vertices. A graph is chromatically unique if it is determined by its chromatic polynomial, up to isomorphism. In other words, G is chromatically unique, then would imply that G and H are isomorphic. All cycle graphs are chromatically unique. Chromatic roots A root (or zero) of a chromatic polynomial, called a “chromatic root”, is a value x where . Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs, for x ≥ 4. This would have established the four color theorem. No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root of every graph with at least one edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27. A result of Tutte connects the golden ratio with the study of chromatic roots, showing that chromatic roots exist very close to : If is a planar triangulation of a sphere then While the real line thus has large parts that contain no chromatic roots for any graph, every point in the complex plane is arbitrarily close to a chromatic root in the sense