Graphs, partitions and Fibonacci numbers

Abstract

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 n ) · n 3 / 4 for constants A , B as n → ∞ . This is proved by using a natural correspondence between partitions of integers and star-like trees.