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One deals with r-regular bipartite graphs with 2n vertices. In a previouspaper Butera, Pernici, and the author have introduced a quantity d(i), afunction of the number of i-matchings, and conjectured that as n goes toinfinity 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 graphpositivity" is the conjecture that for each i and k the probability thatDelta^k (d(i) is non-negative goes to 1 as n goes to infinity. Here we provethis for the range of parameters where r < 11, i+k < 101, k < 21, or i+k < 30all r. A formalism of Wanless as systematized by Pernici is central to thiseffort.

A PROOF of Weak Graph Positivity, for a Large Range of the Parameters