Calculate chromatic number from chromatic polynomial

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I was wondering if there is a way to calculate the chromatic number of a graph knowing only the chromatic polynomial, but not the actual graph. Could someone help me?

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1 Answer

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From the wikipedia page for Chromatic Polynomials:

The chromatic polynomial includes at least as much information about the colorability of G as does the chromatic number. Indeed, the chromatic number is the smallest positive integer that is not a zero of the chromatic polynomial,$$ \chi_G = \min \{k \in \mathbb N ~|~ P_G(k) > 0 \} $$

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I hope this helps ^_^

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