Random Regular Bipartite Graphs Satisfy Weak Virial Positivity, for a Large Range of the Parameters

Abstract

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 "Weak Virial Positivity Conjecture" is the conjecture that for each $i$ and $k > 1$ the probability that $Δ^k u(i) > 0$ goes to $1$ as $n$ goes to infinity. The term Virial is used since the condition $Δ^k u(i) > 0$ corresponds to the positivity of the Virial coefficients for infinite regular lattices. Herein we prove Weak Virial Positivity for the range of parameters $r < 11$, $i+k < 101$, $1 < k < 28$,or $i+k < 30$ all $r$. A formalism of Wanless as systematized by Pernici is central to this effort. Basically this paper is a corollary to our parallel attack on graph positivity in a previous paper. We assume basic knowledge of this previous paper.