Ultra log-concavity and real-rootedness of dependence polynomials

For some positive integer m, a real polynomial P (x) =₊=₀ᵐaₖxᵏ with aₖ 0 is called log-concave (resp. ultra log-concave) if aₖ² a₊-₁a₊+₁ (resp. aₖ² (1+1k) (1+1m-k) a₊-₁a₊+₁) for all 1 k m-1. If P (x) has only real roots, then it is called real-rooted. It is well-known that the conditions of log-concavity, ultra log-concavity and real-rootedness are ever-stronger. For a graph G, a dependent set is a set of vertices which is not independent, i. e. , the set of vertices whose induced subgraph contains at least one edge. The dependence polynomial of G is defined as D (G, x): =₊ ₀dₖ (G) xᵏ, where dₖ (G) is the number of dependent sets of size k in G. Horrocks proved that D (G, x) is log-concave for every graph G J. Combin. Theory, Ser. B, 84 (2002) 180--185. In the present paper, we prove that, for a graph G, D (G, x) is ultra log-concave if G is (K₂ 2K₁) -free or contains an independent set of size |V (G) |-2, and give the characterization of graphs whose dependence polynomials are real-rooted. Finally, we focus more attention to the problems of log-concavity about independence systems and pose several conjectures closely related the famous Mason's Conjecture.