On clique polynomials

Let G be a simple graph. We assign a polynomial C(G; x) to G, called the clique polynomial, where the coefficient of Xi, i > 0, is the number of cliques of G with i vertices, and the constant term is 1. Fisher and Solow (1990), proved that this polynomial always has a real root. We prove this result by a simple and elementary method, which also implies the following results. If (G is the greatest real root of C (G; x) then for an induced subgraph H of G, (H ::; (G, and for a spanning subgarph H of G, (H ~ (G. As a consequence of the first inequality we have a(G) :::; -I/(G, where a(G) denotes the independence number of G.