Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers

Öteleş, Ahmet

doi:10.2478/auom-2019-0022

Abstract

In this paper, we consider the relationships between the numbers of perfect matchings (1-factors) of bipartite graphs and Pell, Mersenne and Perrin Numbers. Then we give some Maple procedures in order to calculate the numbers of perfect matchings of these bipartite graphs.

References

[1] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.10.1002/9781118033067
Search in Google Scholar Back to article
[2] T. Koshy, Fibonacci, Lucas, and Pell numbers, and Pascal’s triangle, Math. Spectrum, 43(3) (2011), 125-132.
Search in Google Scholar Back to article
[3] A.F. Horadam, Jacobstal and Pell Curves, Fibonacci Quart., 26 (1988), 79-83.10.1080/00150517.1988.12429664
Search in Google Scholar Back to article
[4] The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, https://oeis.or, 2013.
Search in Google Scholar Back to article
[5] P. Catarino, H. Campos, P. Vasco, On the Mersenne sequence, Annales Mathematicae et Informaticae, 46 (2016), 37-53.
Search in Google Scholar Back to article
[6] W. Adams, D. Shanks, Strong primality tests that are not sufficient, Mathematics of Computation, 39(159) (1982), 255-300.10.1090/S0025-5718-1982-0658231-9
Search in Google Scholar Back to article
[7] A.S. Asratian, T.M.J. Denley, R. Häggkvist, Bipartite Graphs and their Applications, Cambridge Tracts in Mathematics, 131, Cambridge University Press, 1998.10.1017/CBO9780511984068
Search in Google Scholar Back to article
[8] H. Minc, Permanents, Encyclopedia of mathematics and its applications, Addison-Wesley, New York, 1978.
Search in Google Scholar Back to article
[9] G.W. Wheland, The Theory of Resonant and its Application to Organic Chemistry, Wiley, New York, 1953.
Search in Google Scholar Back to article
[10] R.A. Brualdi, P.M. Gibson, Convex polyhedra of doubly stochastic matrices I: applications of the permanent function, J. Combin. Theory A, 22 (1977), 194-230.10.1016/0097-3165(77)90051-6
Search in Google Scholar Back to article
[11] R.A. Brualdi, D. Cvetkovic, A Combinatorial Approach to Matrix Theory and Its Applications, CRC Press, 2009.10.1201/9781420082241
Search in Google Scholar Back to article
[12] F. Harary, Determinants, permanents and bipartite graphs, Math. Mag., 42 (1969), 146-148.10.1080/0025570X.1969.11975950
Search in Google Scholar Back to article
[13] M. Marcus, H. Minc, Permanents, Amer. Math. Monthly, 72 (1965), 577-591.10.1080/00029890.1965.11970575
Search in Google Scholar Back to article
[14] G.Y. Lee, S.G. Lee, H. G. Shin, On the k-generalized Fibonacci matrix Q_k, Lin. Alg. Appl., 251 (1997), 73-88.10.1016/0024-3795(95)00553-6
Search in Google Scholar Back to article
[15] G.Y. Lee, k-Lucas numbers and associated bipartite graphs, Lin. Alg. Appl., 320 (2000), 51-61.10.1016/S0024-3795(00)00204-4
Search in Google Scholar Back to article
[16] W.C. Shiu, Peter C.B. Lam, More on the generalized Fibonacci numbers and associated bipartite graphs, Int. Math. J., 3 (2003), 5-9.
Search in Google Scholar Back to article
[17] E. Kılıç, D. Tasçı, On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers, Ars Combin., 89 (2008), 31-40.
Search in Google Scholar Back to article
[18] G.Y. Lee, S.G. Lee, A note on generalized Fibonacci numbers, Fibonacci Quart., 33 (1995), 273-278.10.1080/00150517.1995.12429146
Search in Google Scholar Back to article
[19] E. Kılıç, D. Taşçı, On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers, Rocky Mt. J. Math., 37(6) (2007), 1953-1969.10.1216/rmjm/1199649832
Search in Google Scholar Back to article
[20] M. Akbulak, A. Öteleş, On the number of 1-factors of bipartite graphs, Math. Sci. Lett., 2(3) (2013), 1-7.10.12785/msl/020306
Search in Google Scholar Back to article
[21] A. Öteleş, On the Number of Perfect Matchings for Some Certain Types of Bipartite Graphs, Filomat, 31(15) (2017), 48094818.10.2298/FIL1715809O
Search in Google Scholar Back to article
[22] F. Yilmaz and D. Bozkurt, Some properties of Padovan sequence by matrix methods, Ars Combin., 104 (2012), 149-160.
Search in Google Scholar Back to article
[23] F. Yilmaz and D. Bozkurt, Hessenberg matrices and the Pell and Perrin numbers, J. Number Theory, 131 (2011), 1390-1396.10.1016/j.jnt.2011.02.002
Search in Google Scholar Back to article
[24] C.M. da Fonseca, T. Sogabe and F. Yilmaz, Lower k-Hessenberg Matrices and k-Fibonacci, Fibonacci-p and Pell (p,i) Number, Gen. Math. Notes, 31(1) (2015), 10-17.
Search in Google Scholar Back to article
[25] E. Kılıç, D. Tasçı, On families of bipartite graphs associated with sums of generalized order-k Fibonacci and Lucas numbers, Ars Combin., 94 (2008), 13-23.
Search in Google Scholar Back to article
[26] E. Kılıç, A.P. Stakhov, On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices, Chaos Solitons Fractals, 40(22) (2009), 10-21.10.1016/j.chaos.2007.10.007
Search in Google Scholar Back to article

Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers

Abstract

Paradigm

My account