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)=∑nk=0mk(G)λk. In this paper we show that if G is a d--regular graph and 0<λ<(4d)−2, then 1v(G)lnMG(λ)>1v(Kd+1)lnMKd+1(λ). The same inequality holds true if d=3 and λ<0.3575. More precise conjectures are also given.