  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) for all k and i,
approaches 1. Here Delta is the finite difference operator. This conjecture we
called the 'graph positivity conjecture'.
  In this paper it is formally shown that for each i and k the probability that
Delta^k d(i) goes to 1 with n going to infinity.
  We call this weaker result the 'weak graph positivity conjecture ( theorem
)'.
  A formalism of Wanless as systematized by Pernici is central to this effort.
  Our result falls short of being a rigorous proof since we make a sweeping
conjecture ( computer tested ), of which we so far have only a portion of the
proof.
