A PROOF of Weak Graph Positivity, for a Large Range of the Parameters
Paul Federbush
备注：
18 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
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.
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 con…
We deal with $r$-regular bipartite graphs with $2n$ vertices. In a previous paper, Butera, Pernici and the author have introduced a quantity $u(i)$, $u(i) = -\ln(i!m\_i)$, a function of the number of $i$-matchings, $m\_i$, and conjectured that the fraction of graphs that violate $Δ^k u(i) > 0$ for $k > 1$ vanishes as $n$ goes to infinity. Here $Δ$ is the finite difference operator. We now more particularly define the "Virial Positivity Conjecture" as the conjecture that the fraction of graphs that satisfy $Δ^k u(i)$ go to 0 for all $k > 1$ and $i$, approaches 1 as $n$ goes to infinity. The "We…
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