Mason's Conjecture asserts that for an m-element rank r matroid M the sequence I k / m k : 0 ≤ k ≤ r is logarithmically concave, in which I k is the number of independent k-sets of M. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of M satisfies a strong negative correlation property we call… (More)
Abstmct-We reveal an equivalence relation between the construction of a new class of low density MDS array codes, that we call B-Code, and a com-binatorial problem known as perfect one-factorization of complete graphs. We use known perfect one-factors of complete graphs to create constructions and decoding algorithms for both B-Code and its dual code.… (More)
For a nite multigraph G, the reliability function of G is the probability R G (q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if q 2 C is such that R… (More)
In 1981, Stanley applied the Aleksandrov–Fenchel Inequalities to prove a logarithmic concavity theorem for regular matroids. Using ideas from electrical network theory we prove a generalization of this for the wider class of matroids with the " half–plane property ". Then we explore a nest of inequalities for weighted basis– generating polynomials that are… (More)
A Rayleigh matroid is one which satisfies a set of inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. We show that every matroid of rank three satisfies these inequalities.
Let A = a ij be a real n n matrix with non-negative e n tries which are weakly increasing down columns. If B = b ij is the nn matrix where b ij := a ij +z; then we conjecture that all of the roots of the permanent o f B, as a polynomial i n z; are real. Here we establish several special cases of the conjecture.
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial k m k (G)y k has only real nonpositive zeros, in which m k (G) is the number of k-edge matchings of a graph G. There is a stronger multivariate version of this theorem. We provide a general method by which " theorems of Heilmann-Lieb type " can be proved for a wide… (More)
For a finite multigraph G, let Λ(G) denote the lattice of integer flows of G – this is a finitely generated free abelian group with an integer-valued positive definite bilinear form. Bacher, de la Harpe, and Nagnibeda show that if G and H are 2-isomorphic graphs then Λ(G) and Λ(H) are isometric, and remark that they were unable to find a pair of… (More)