ON THE INDEPENDENCE POLYNOMIAL OF UNICYCLIC GRAPHS

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) = P n k=0 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. In this paper we study the independence polynomial of unicyclic graphs( a unicyclic graph is a graph containing exactly one cycle). 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, where C n is the cycle on n vertices.