A Near Proof of Weak Graph Positivity, A New Property of Random Regular Graphs

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.