Abstract
The Fibonacci number of a graph, defined by Prodinger and Tichy in 1982, is the number of independent sets on the graph. The Fibonacci number of the path graph, P n , is the Fibonacci number F n+2 and the Fibonacci number of the cycle graph, C n , is the Lucas number L n . This paper combines the visual nature of graph theory with combinatorial methods to prove new identities for the Fibonacci and Lucas numbers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Benjamin, A.T., Quinn, J.J.: Proofs That Really Count: The Art of Combinatorial Proof. The Mathematical Association of America, Washington, DC (2003)
Calkin, N.J., Wilf, H.S.: The number of independent sets in a grid graph. SIAM J. Discrete Math. 11(1), 54–60 (1998, electronic)
Engel, K.: On the Fibonacci number of an m × n lattice. Fibonacci Quart. 28(1), 72–78 (1990)
Harary, F., Read, R.: Is the null graph a pointless concept? In: Graphs and Combinatorics Conference, George Washington University. Springer, New York (1973)
Knopfmacher, A., Tichy, R.F., Wagner, S., Ziegler, V.: Graphs, partitions, and Fibonacci numbers. Discrete Appl. Math. 155, 1175–1187 (2007)
Koshy, T.: Fibonacci and Lucas numbers with applications. Wiley, New York (2001)
Nelsen, R.B.: Proofs Without Words: Exercises in Visual Thinking. The Mathematical Association of America, Washington, DC (1993)
Nelsen, R.B.: Proofs Without Words. II: More Exercises in Visual Thinking. The Mathematical Association of America, Washington, DC (2001)
Prodinger, H., Tichy, R.: Fibonacci numbers of graphs. Fibonacci Quart. 20(1),16–21 (1982)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
DeMaio, J., Jacobson, J. (2013). Fibonacci and Lucas Identities via Graphs. In: Rychtář, J., Gupta, S., Shivaji, R., Chhetri, M. (eds) Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference. Springer Proceedings in Mathematics & Statistics, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9332-7_9
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9332-7_9
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9331-0
Online ISBN: 978-1-4614-9332-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)