Beraha Constants
المؤلف:
Beraha, S. Ph.D
المصدر:
thesis. Baltimore, MD: Johns Hopkins University, 1974.
الجزء والصفحة:
...
19-2-2020
1037
Beraha Constants
The 
th Beraha constant (or number) is given by
is 
, where 
is the golden ratio, 
is the silver constant, and 
. The following table summarizes the first few Beraha numbers.
 |
 |
approx. |
| 1 |
4 |
|
| 2 |
0 |
|
| 3 |
1 |
|
| 4 |
2 |
|
| 5 |
 |
2.618 |
| 6 |
3 |
|
| 7 |
 |
3.247 |
| 8 |
 |
3.414 |
| 9 |
 |
3.532 |
| 10 |
 |
3.618 |
Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of 
(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).
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 
." Research Report COPR 72-7, University of Waterloo, 1972a.
Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case 
." Research Report COPR 72-4, University of Waterloo, 1972b.
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