21-37 Paul Federbush

Random Regular Bipartite Graphs Satisfy Weak Virial Positivity, for a Large Range of the Parameters (13K, LaTeX) Jul 9, 21

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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. 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

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