Ultra log-concavity and real-rootedness of dependence polynomials

For some positive integer m, a real polynomial P(x)=∑ₖ₌₀ᵐaₖxᵏ with aₖ 0 is called log-concave (resp. ultra log-concave) if aₖ² aₖ₋₁aₖ₊₁ (resp. aₖ² (1+1/k)(1+1/m-k)· aₖ₋₁aₖ₊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):=∑k 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.