A PROOF of Weak Graph Positivity, for a Large Range of the Parameters
arXiv:2012.10927
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)) 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.
arXiv comment18 pages, basic corrections to Appendix B, some computer proofs, appendix with work of Robin Chapman. added range of validity from new paper on mp_arc but not arXiv. arXiv admin note: text overlap with arXiv:1710.00357