TOPICS
Search

Beraha Constants


The nth Beraha constant (or number) is given by

 B(n)=2+2cos((2pi)/n).

B(5) is phi+1, where phi is the golden ratio, B(7) is the silver constant, and B(10)=phi+2. The following table summarizes the first few Beraha numbers.

nB(n)approx.
14
20
31
42
51/2(3+sqrt(5))2.618
63
72+2cos(2/7pi)3.247
82+sqrt(2)3.414
92+2cos(2/9pi)3.532
101/2(5+sqrt(5))3.618

Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of B_(10) (G. Royle, pers. comm., Nov. 21, 2005). However, the roots of chromatic polynomials of planar triangulations appear to cluster around the Beraha numbers (and, technically, are conjectured to be accumulation points of roots of planar triangulation chromatic polynomials).


See also

Chromatic Polynomial, Golden Ratio, Silver Constant

Explore with Wolfram|Alpha

References

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 lambda=1." Research Report COPR 72-7, University of Waterloo, 1972a.Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case lambda=infty." Research Report COPR 72-4, University of Waterloo, 1972b.

Referenced on Wolfram|Alpha

Beraha Constants

Cite this as:

Weisstein, Eric W. "Beraha Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BerahaConstants.html

Subject classifications