On bipartite graphs and the Fibonacci numbers

Fatih Yılmaz and Pınar Eldutar
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 143–149
DOI: 10.7546/nntdm.2019.25.4.143-149
Full paper (PDF, 157 Kb)

Details

Authors and affiliations

Fatih Yılmaz
Department of Mathematics, Polatlı Art and Science Faculty,
Ankara Haci Bayram Veli University, 06900 Ankara, Turkey

Pınar Eldutar
Department of Mathematics,
Gazi University, 06900 Ankara, Turkey

Abstract

In this short note, we consider adjacency matrices of ladder graphs. Then we obtain permanental polynomials, eigenvalues and some other properties of adjacency matrix of the graph.

Keywords

• Permanental polynomial
• Fibonacci numbers
• Eigenvalue

2010 Mathematics Subject Classification

• 15A15
• 11B39

References

1. Borowiecki, M., & Jozwiak, T. (1982). Computing the permanental polynomial of a multigraph, Discuss. Math., 5, 9–16.
2. Cahil, N. D., D’Errico, J. R., & Spence, J. P. (2003). Complex factorizations of the Fibonacci and Lucas numbers, The Fibonacci Quarterly, 41 (1), 13–19.
3. Codenotti, B., & Resta, G. (2002). Computation of spare circulant permanents via
   determinants, Linear Alg. Appl., 355, 15–34.
4. Cvetkocic, D. M., Doob, M., Gutman, I., & Torgasev, A. (1988). Recent Results in the Theory of Graph Spectra, North-Holland, Amsterdam, p. 123.
5. Eldutar, P., & Yilmaz, F. (2016). On linear algebra of adjacency matrices for one type of graph, ICAMA-2016, Ankara, Turkey.
6. Hosoya, H. (2004). Sequences of polyomino and polyhex graphs whose perfect matching numbers are Fibonacci or Lucas numbers: The Golden family graphs of a new category, Forma, 19, 405–412.
7. Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods, The Fibonacci Quarterly, 20 (1), 73–76.
8. Kasum, D., Trinajstic, N., & Gutman, I. (1981). Chemical graph theory. III, On the permanental polynomial, Croat. Chem. Acta, 54, 321–328.
9. Merris, R., Rebman K. R., & Watkins, W. (1981). Permanental polynomials of graphs, Lin. Algebra Appl, 38, 273–288.
10. Rimas, J. (2006). On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order–II, Applied Mathematics and Computation, 172, 245–251.
11. Rosenfeld, V. R., & Gutman, I. (1989). A novel approach to graph polynomials, Match-Commun. Math. Comput. Chem., 24, 191–199.
12. Schwenk, A. J. (1974). Computing the Characteristic Polynomial of a Graph, Springer–Verlag, Berlin, pp. 153–172.
13. Shannon, A., Turner, J., & Atanassov, K. (1991). A generalized tableau associated with colored convolution trees, Discrete Mathematics, 92 (1–3), 329–340.
14. Strang, G. (2009). Introduction to Linear Algebra, Wellesley-Cambridge Press.
15. Turner, J. (1968). Generalized matrix functions and the graph isomorphism problem, SIAM J. Appl. Math., 16, 520–526.
16. Vasudev, C. (2006). Graph Theory with Applications, New Age International Limited Publishers, New Delhi.
17. Zhang, H., & Ding, F. (2013). On the Kronecker products and their applications, Hindawi Publishing Corporation Journal of Applied Mathematics, Article ID 296185.