MATCHINGS IN REGULAR GRAPHS: MINIMIZING THE PARTITION FUNCTION

For a graph G on v(G) vertices let m(k)(G) denote the number of matchings of size k, and consider the partition function M-G(lambda) = Sigma(n)(k=0)m(k)(G)lambda(k). In this paper we show that if G is a d-regular graph and 0 < lambda < (4d)(-2), then 1/v(G) ln M-G(lambda) > 1/v(Kd+1) ln MKd+1(lambda). The same inequality holds true if d = 3 and lambda < 0.3575. More precise conjectures are also given.