Lucas Sayılarıyla İlişkili İki Parçalı Graflar
Year 2020,
Volume: 13 Issue: 3, 1069 - 1076, 31.12.2020
Ahmet Öteleş
Diyar Omar Mustafa Zangana
Abstract
Bu çalışmada iki parçalı komşuluk matrisi n×n mertebeli (0,1)-circulant matrisi olan iki parçalı bir graf ele aldık. Daha sonra bu grafın mükemmel eşlemelerinin (1-factor) sayılarının Lucas sayıları ile arasındaki ilişkiyi verdik. Son olarak da bu mükemmel eşlemelerin sayısını hesaplamak için bazı maple prosedürleri verdik.
References
- Asratian, A. S., Denley, T. M. J. and Häggkvist, R. 1998. "Bipartite Graphs and their Applications", Cambridge Tracts in Mathematics, 131, Cambridge University Press.
- Brualdi, R. A. and Gibson, P. M. 1977. "Convex polyhedra of doubly stochastic matrices I: applications of the permanent function", J. Combin. Theory, A 22, 194-230.
- Brualdi, R. A. and Cvetkovic, D. 2009. "A Combinatorial Approach to Matrix Theory and Its Applications", CRC Press.
- Fonseca, C. M. da, Sogabe, T. and Yilmaz, F. 2015. "Lower k-Hessenberg Matrices and k-Fibonacci, Fibonacci-p and Pell (p,i) Number", General Mathematics Notes, 31(1), 10-17.
- Harary, F. 1969. "Determinants, permanents and bipartite graphs", Mathematics Magazine, 42, 146-148.
- Kılıç, E. and Tasçı, D. 2007. "On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers", Rocky Mountain Journal of Mathematics,
37(6), 1953-1969.
- Kılıç, E. and Tasçı, D. 2008. "On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers", Ars Combinatoria, 89, 31-40.
- Kılıç, E. and Stakhov, A. P. 2009. "On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices", Chaos, Solitons&Fractals, 40(22), 10-21.
- Koshy, T. 2001. "Fibonacci and Lucas Numbers with Applications", Wiley-Interscience, New York.
- Koshy, T. 2011. "Fibonacci, Lucas, and Pell numbers, and Pascal's triangle", Mathematical Spectrum, 43(3), 125-132.
- König, D. 1915. "Vonalrendszerek és determinások", Math. Termész. Ért., 33, 221-229.
- König, D. 1916. "Über Graphen und ihre Anwendungen", Math. Annelen, 77, 453-465.
- Lee, G. Y. and Lee, S. G. 1995. "A note on generalized Fibonacci numbers", The Fibonacci Quarterly, 33, 273-278.
- Lee, G. Y., Lee, S. G. and Shin, H. G. 1997. "On the k-generalized Fibonacci matrix Q_{k}", Linear Algebra and its Applications, 251, 73-88.
- Lee, G. Y. 2000. "k-Lucas numbers and associated bipartite graphs", Linear Algebra and its Applications, 320, 51-61.
- Marcus, M. and Minc, H. 1965. "Permanents", American Mathematical Monthly, 72, 577-591.
- Minc, H. 1978. "Permanents, Encyclopedia of mathematics and its applications", Addison-Wesley, New York.
- Öteleş, A., 2017. "On the number of perfect matchings for some certain types of bipartite graphs", Filomat, 31(5), 4809-4818.
- Özkan, E. and Altun, İ, M. 2019. "Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials", Communications in Algebra, 47(10), 4020-4030.
- Özkan, E., Taştan, M. and Aydoğdu, A. 2018. "2-Fibonacci polynomials in the family of Fibonacci numbers", Notes on Number Theory and Discrete Mathematics, 24(3), 47-55.
- Özkan, E. and Taştan, M. 2019. "k-Fibonacci Polynomials in The Family of Fibonacci Numbers", Research Reviews: Discrete Mathematical Structures, 6(3), 19-22.
- Shiu, W. C. and Lam, P. C. B. 2003. "More on the generalized Fibonacci numbers and associated bipartite graphs", Int. Math., J., 3, 5-9.
- The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org (Son Erişim Tarihi: 05.03.2020)
- Wheland, G. W. 1953. "The Theory of Resonant and its Application to Organic Chemistry", Wiley, New York.
- Yilmaz, F. and Bozkurt, D. 2012. "Some properties of Padovan sequence by matrix methods", Ars Combinatoria, 104, 149-160.
Bipartite Graphs Associated with Lucas Numbers
Year 2020,
Volume: 13 Issue: 3, 1069 - 1076, 31.12.2020
Ahmet Öteleş
Diyar Omar Mustafa Zangana
Abstract
In this paper, we consider the bipartite graph whose bipartite adjacency matrix is an n×n (0,1)-circulant matrix. Then we show that the numbers of perfect matchings of this graph are equal to the well-known Lucas numbers. Finally, we give some Maple procedures in order to calculate the numbers of perfect matchings of the bipartite graph.
References
- Asratian, A. S., Denley, T. M. J. and Häggkvist, R. 1998. "Bipartite Graphs and their Applications", Cambridge Tracts in Mathematics, 131, Cambridge University Press.
- Brualdi, R. A. and Gibson, P. M. 1977. "Convex polyhedra of doubly stochastic matrices I: applications of the permanent function", J. Combin. Theory, A 22, 194-230.
- Brualdi, R. A. and Cvetkovic, D. 2009. "A Combinatorial Approach to Matrix Theory and Its Applications", CRC Press.
- Fonseca, C. M. da, Sogabe, T. and Yilmaz, F. 2015. "Lower k-Hessenberg Matrices and k-Fibonacci, Fibonacci-p and Pell (p,i) Number", General Mathematics Notes, 31(1), 10-17.
- Harary, F. 1969. "Determinants, permanents and bipartite graphs", Mathematics Magazine, 42, 146-148.
- Kılıç, E. and Tasçı, D. 2007. "On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers", Rocky Mountain Journal of Mathematics,
37(6), 1953-1969.
- Kılıç, E. and Tasçı, D. 2008. "On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers", Ars Combinatoria, 89, 31-40.
- Kılıç, E. and Stakhov, A. P. 2009. "On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices", Chaos, Solitons&Fractals, 40(22), 10-21.
- Koshy, T. 2001. "Fibonacci and Lucas Numbers with Applications", Wiley-Interscience, New York.
- Koshy, T. 2011. "Fibonacci, Lucas, and Pell numbers, and Pascal's triangle", Mathematical Spectrum, 43(3), 125-132.
- König, D. 1915. "Vonalrendszerek és determinások", Math. Termész. Ért., 33, 221-229.
- König, D. 1916. "Über Graphen und ihre Anwendungen", Math. Annelen, 77, 453-465.
- Lee, G. Y. and Lee, S. G. 1995. "A note on generalized Fibonacci numbers", The Fibonacci Quarterly, 33, 273-278.
- Lee, G. Y., Lee, S. G. and Shin, H. G. 1997. "On the k-generalized Fibonacci matrix Q_{k}", Linear Algebra and its Applications, 251, 73-88.
- Lee, G. Y. 2000. "k-Lucas numbers and associated bipartite graphs", Linear Algebra and its Applications, 320, 51-61.
- Marcus, M. and Minc, H. 1965. "Permanents", American Mathematical Monthly, 72, 577-591.
- Minc, H. 1978. "Permanents, Encyclopedia of mathematics and its applications", Addison-Wesley, New York.
- Öteleş, A., 2017. "On the number of perfect matchings for some certain types of bipartite graphs", Filomat, 31(5), 4809-4818.
- Özkan, E. and Altun, İ, M. 2019. "Generalized Lucas polynomials and relationships between the Fibonacci polynomials and Lucas polynomials", Communications in Algebra, 47(10), 4020-4030.
- Özkan, E., Taştan, M. and Aydoğdu, A. 2018. "2-Fibonacci polynomials in the family of Fibonacci numbers", Notes on Number Theory and Discrete Mathematics, 24(3), 47-55.
- Özkan, E. and Taştan, M. 2019. "k-Fibonacci Polynomials in The Family of Fibonacci Numbers", Research Reviews: Discrete Mathematical Structures, 6(3), 19-22.
- Shiu, W. C. and Lam, P. C. B. 2003. "More on the generalized Fibonacci numbers and associated bipartite graphs", Int. Math., J., 3, 5-9.
- The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org (Son Erişim Tarihi: 05.03.2020)
- Wheland, G. W. 1953. "The Theory of Resonant and its Application to Organic Chemistry", Wiley, New York.
- Yilmaz, F. and Bozkurt, D. 2012. "Some properties of Padovan sequence by matrix methods", Ars Combinatoria, 104, 149-160.