Let G be a simple graph on n vertices. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I ( G , x ) = ∑ k = 0 n s ( G , k ) x k , where s ( G, k ) is the number of independent sets of G with size k and s ( G , 0) = 1. A unicyclic graph is a graph containing exactly one cycle. Let C_(n) be the cycle on n vertices. In this paper we study the independence polynomial of unicyclic graphs. We show that among all connected unicyclic graphs G on n vertices (except two of them), I ( G, t ) > I ( C_(n), t ) for sufficiently large t . Finally for every n ≥ 3 we find all connected graphs H such that I ( H, x ) = I ( C_(n), x ).
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