On Families Of Bipartite Graphs Associated With Sums Of Generalized Order-k Fibonacci And Lucas Numbers

In this paper, we consider the relationships between the sums of the generalized order-k Fibonacci and Lucas numbers and 1-factors of bipartite graphs. 1. Introduction We consider the generalized order k Fibonacci and Lucas numbers. In [1], Er de…ned k sequences of the generalized order k Fibonacci numbers as shown: g n = k X j=1 g n j ; for n > 0 and 1 i k; (1.1) with boundary conditions for 1 k n 0; g n = 1 if i = 1 n; 0 otherwise, where g n is the nth term of the ith sequence. For example, if k = 2, then g n is usual Fibonacci sequence, fFng ; and, if k = 4, then the 4th sequence of the generalized order 4 Fibonacci numbers is 1; 1; 2; 4; 8; 15; 29; 56; 108; 208; 401; 773; 1490; : : : : In [9], the authors de…ned k sequences of the generalized order k Lucas numbers as shown: l n = k X j=1 l n j , for n > 0 and 1 i k, (1.2) 2000 Mathematics Subject Classi…cation. 11B39, 15A15, 15A36, 05C50.