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\(k\)-Fibonacci numbers and \(k\)-Lucas numbers and associated bipartite graphs. (English) Zbl 1496.11026

Summary: In [E. Kilic, J. Comput. Appl. Math. 209, No. 2, 133–145 (2007; Zbl 1162.11013); G.-Y. Lee et al., Discrete Appl. Math. 130, No. 3, 527–534 (2003; Zbl 1020.05016); G.-Y. Lee and S.-G. Lee, Fibonacci Q. 33, No. 3, 273–278 (1995; Zbl 0834.11009)], several authors studied the generalized Fibonacci numbers. Also, in [Linear Algebra Appl. 320, No. 1–3, 51–61 (2000; Zbl 0960.05079)], the author found a class of bipartite graphs whose number of 1-factors is the \(n\)th \(k\)-Lucas numbers. In this paper, we give a new relationship between \(g_n^{(k)}\) and \(l_n^{(k)}\) and the number of 1-factors of a bipartite graph.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A19 Combinatorial identities, bijective combinatorics
15A15 Determinants, permanents, traces, other special matrix functions
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