  One deals with r-regular bipartite graphs with 2n vertices. In a previous
paper Butera, Pernici, and the author have introduced a quantity d(i), a
function of the number of i-matchings, and conjectured that as n goes to
infinity the fraction of graphs that satisfy Delta^k( d(i)) is non-negative,
for all k and i, approaches 1. Here Delta is the finite difference operator.
This conjecture we called the "graph positivity conjecture". "Weak graph
positivity" is the conjecture that for each i and k the probability that
Delta^k (d(i) is non-negative goes to 1 as n goes to infinity. Here we prove
this for the range of parameters where r < 11, i+k < 101, k < 21, or i+k < 30
all r. A formalism of Wanless as systematized by Pernici is central to this
effort.
