matchings in regular graphs: minimizing the partition function

For a graph g on v(g) vertices let mk(g) denote the number of matchings of size k, and consider the partition function mg(λ) = ∑^n k=0^ mk(g)λ^k . in this paper we show that if g is a d–regular graph and 0 < λ < (4d)^−2 , then 1/ v(g) ln mg(λ) > 1/v(kd+1) ln mkd+1 (λ). the same inequality holds true if d = 3 and λ < 0.3575. more precise conjectures are also given.