Combinatorics Seminar

We deal always with r-regular bipartite graphs or lattices. We state two results discussed in the talk. First, for the lattice gas of dimers on the hyper-rectangular lattice of any dimension, the first 20 terms in the virial expansion are positive. Second, for a graph of v vertices we define a function d(i) of the number of i-matchings by ln( N/r^i)-ln(N'/(v-1)^i), where N is the number of i-matchings of the graph, N' the number of i-matchings of its completion. We define delta by delta f(z) = f(z+1)- f(z). Then, if j+k < 30, the fraction of graphs with v vertices that satisfy delta^k f(j) > 0 approaches 1 as v goes to infinity.
Speaker(s): Paul Federbush (University of Michigan)