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Graph-theoretic models for the Fibonacci family

Published online by Cambridge University Press:  23 January 2015

Thomas Koshy
Thomas Koshy*
Affiliation:
Framingham State University, Framingham. MA 01701-9101, USA
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Extract

The well-known Fibonacci and Lucas numbers continue to faxcinate the mathematical community with their beauty, elegance, ubiquity, and applicability. After several centuries of exploration, they are still a fertile ground for additional activities, for Fibonacci enthusiasts and amateurs alike.

Fibonacci numbers Fn and Lucas numbers Ln belong to a large integer family {xn}, often defined by the recurrence xn = xn−1 + xn−2, where x1 = a, x2 = b, and n ≥ 3. When a = b = 1, xn = Fn; and when a = 1 and b = 3, xn = Ln. Clearly, F0 = 0 and L0 = 2. They satisfy a myriad of elegant properties [1,2,3]. Some of them are:

In this article, we will give a brief introduction to the Q-matrix, employ it in the construction of graph-theoretic models [4, 5], and then explore some of these identities using them.

In 1960 C.H. King studied the Q-matrix

Type
Articles
Information
The Mathematical Gazette , Volume 98 , Issue 542 , July 2014 , pp. 256 - 265
Copyright
Copyright © Mathematical Association 2014 

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References

1. Benjamin, A. T. and Quinn, J. J., Proofs that really count, Mathematical Association of America, Washington, D.C., 2003.Google Scholar
2. Brigham, R. C. et al, A tiling scheme for the Fibonacci Numbers, Journal of Recreational Mathematics, 28 (1996-1997), pp. 10–16.Google Scholar
3. Koshy, T., Fibonacci and Lucas numbers with applications, Wiley, New York, 2001.Google Scholar
4. Huang, D., Fibonacci identities, matrices, and graphs, Mathematics Teacher 98 (2005), pp. 400–403.Google Scholar
5. Krebs, M. and Martinez, N.C., The combinatorial trace method in action, The College Mathematics Journal 44 (2013), pp. 32–36.Google Scholar
6. Koshy, T., Discrete Mathematics with Applications, Elsevier Academic Press, Burlington, Massachusetts (2004).Google Scholar