Beraha constants

The Beraha constants are a series of mathematical constants by which the [math]\displaystyle{ n\text{th} }[/math] Beraha constant is given by

[math]\displaystyle{ B (n) = 2 + 2 \cos \left ( \frac{2\pi}{n} \right ). }[/math]

Notable examples of Beraha constants include [math]\displaystyle{ B (5) }[/math]is [math]\displaystyle{ \varphi + 1 }[/math], where [math]\displaystyle{ \varphi }[/math] is the golden ratio, [math]\displaystyle{ B (7) }[/math]is the silver constant^[1] (also known as the silver root),^[2] and [math]\displaystyle{ B (10) = \varphi + 2 }[/math].

The following table summarizes the first ten Beraha constants.

See also

• Chromatic polynomial

Notes

References

• Weisstein, Eric W.. "Beraha Constants". http://mathworld.wolfram.com/BerahaConstants.html. 
• Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.
• Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983.
• Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160–163, 1986.
• Tutte, W. T. "Chromials." University of Waterloo, 1971.
• Tutte, W. T. "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case [math]\displaystyle{ \lambda = 1 }[/math]," Research Report COPR 72–7, University of Waterloo, 1972a.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case [math]\displaystyle{ \lambda = \infty }[/math]." Research Report COPR 72–4, University of Waterloo, 1972b.

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