Properties
For fixed on vertices, the chromatic polynomial is in fact a polynomial; it has degree . Nonisomorphic graphs may have the same chromatic polynomial. By definition, evaluating the chromatic polynomial in yields the number of -colorings of for …. Perhaps surprisingly, the same holds for any, and besides, yields the number of acyclic orientations of . Furthermore, the derivative evaluated at 1, equals the chromatic invariant up to sign.
If has vertices, edges, and components …,, then
- The coefficients of are zeros.
- The coefficients of are all non-zero.
- The coefficient of in is 1.
- The coefficient of in is .
- The coefficients of every chromatic polynomial alternate in signs.
- The absolute values of coefficients of every chromatic polynomial form a log-concave sequence.
A graph G with vertices is a tree if and only if .
Read more about this topic: Chromatic Polynomial
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