Graphs, partitions and Fibonacci numbers

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number > 2(n-1) + 5 have diameter <= 4 and determine the order of these trees with respect to their Fibonacci numbers. Furthermore, it is shown that the average Fibonacci number of a star-like tree (i.e. diameter <= 4) is asymptotically A.2(n).exp(B root n).n(3/4) for constants A, B as n ->infinity. This is proved by using a natural correspondence between partitions of integers and star-like trees. (c) 2006 Elsevier B.V. All rights reserved.