Paul Federbush
 A Near Proof of Weak Graph Positivity, A new Property of Regular Random Groups 
(25K, LaTeX)

ABSTRACT.   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.