Rayleigh Matroids

Abstract

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 [9] in 1992. We prove a variety of results relating Rayleigh matroids to other well-known classes – in particular, we show that a binary matroid is Rayleigh if and only if it does not contain $\mathcal{S}_{8}$ as a minor. This has the consequence that a binary matroid is balanced if and only if it is Rayleigh, and provides the first complete proof in print that $\mathcal{S}_{8}$ is the only minor-minimal binary non-balanced matroid, as claimed in [9]. We also give an example of a balanced matroid which is not Rayleigh.