# Beraha's Conjecture and Cyclic Graphs

Beraha's Conjecture and Cyclic Graphs

The chromatic polynomial of a graph gives the number of ways to color the graph with colors, such that no pair of connected vertices shares the same color.

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Beraha's conjecture (due to Tutte) says that for each , the Beraha number =2+2cos is the limit of roots of at least one family of chromatic polynomials.

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B

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2π

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This Demonstration shows that:

1) All cyclic graphs of odd order have a root of their chromatic polynomial equal to .

B(4)

2) For even order , let be the root with largest real part and positive imaginary part, and let be the complex conjugate root (the chromatic polynomial has real coefficients so complex roots come in conjugate pairs). Then the sequences and converge to as .

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P

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N

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P

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N

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B(4)

n∞