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

TL;DR

This paper proves the weak virial positivity conjecture for large regular bipartite graphs within certain parameter ranges, confirming that most such graphs satisfy positivity conditions related to matchings as the number of vertices grows.

Contribution

It establishes the weak virial positivity conjecture for $r$-regular bipartite graphs with parameters within specified bounds, extending previous conjectures and using a formalism by Wanless.

Findings

Weak virial positivity holds for $r<11$, $i+k<101$, $1<k<28$, or $i+k<30$ for all $r$.

The results support the conjecture that most large regular bipartite graphs satisfy virial positivity conditions.

The proof relies on a formalism systematized by Wanless and builds on previous work on graph positivity.

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 $\Delta^k u(i) > 0$ for $k > 1$ vanishes as $n$ goes to infinity. Here $\Delta$ is the finite difference operator. We now more particularly define the "Virial Positivity Conjecture" as the conjecture that the fraction of graphs that satisfy $\Delta^k u(i)$ 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 $\Delta^k u(i) > 0$ goes to $1$ as $n$ goes to infinity. The term Virial is used since the condition $\Delta^k u(i) > 0$ 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.