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We show in a simple way that for any k,m ∈ N, there exists a tree T such that the number of independent sets of T is congruent to k modulo m. This resolves a conjecture of Wagner (Almost all trees have an even number of independent sets, Electron. J. Combin. 16 (2009), # R93). 1 The number of independent sets in a tree A set of vertices in a graph G is called independent if the set induces no edges. We write i(G) for the number of independent sets in G; i(G) is often known as the Fibonacci number, or in mathematical chemistry as the Merrifield-Simmons index or the σ-index. The study was initiated by Prodinger and Tichy in [4]. In particular, they showed that among trees of the same order, the maximum and minimum Fibonacci numbers are attained by the star and the path respectively. The name stems from the fact that the Fibonacci numbers of paths are the usual Fibonacci numbers. Indeed, as the empty set is independent, i(P0) = 1, i(P1) = 2 and i(Pn) = i(Pn−1) + i(Pn−2) for n � 2. The inverse question asks for a positive integer k, whether there exists a graph G suc

Year: 2010

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