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)
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.
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)
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)
The generalized theta graph s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : :; s k 1. We prove that the roots of the chromatic polynomial ((s 1 ;:::;s k ; z) of a k-ary generalized theta graph all lie in the disc jz ? 1j 1 + o(1)] k= logk, uniformly in the path lengths s i. Moreover, we prove that… (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)
Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. These form a subclass of the balanced matroids defined by Feder and Mihail  in 1992. We prove a variety of results relating Rayleigh ma-troids to other well–known classes – in particular, we show that a binary matroid is Rayleigh if and only if… (More)