The Fibonacci number or -index of a graph is defined as the number of independent vertex subsets. For molecular graphs, this index has several applications in combinatorial chemistry. The talk is based on a joint work of Knopfmacher, Tichy, Ziegler and the speaker. It deals with enumeration problems concerning this graph-theoretical index, such as characterizing graphs of small or large index or calculating the average index of some given graph class. In particular, a special class of trees - the so-called ``star-like'' trees - will be discussed. This class of trees corresponds to partitions of natural numbers in a well-defined way. The main results include the fact that all trees with Fibonacci number (where denotes the number of edges) are star-like and an asymptotic formula for the average Fibonacci number of a star-like tree.

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