A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P_{B(M)} have the half-plane property? Not all matroids have the half-plane property, but we find large classes that do: all sixth-root-of-unity matroids, and a subclass of transversal (or cotransversal) matroids that we call "nice". Furthermore, the class of matroids with the half-plane property is closed under minors, duality, direct sums, 2-sums, series and parallel connection, full-rank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the half-plane property: a determinant construction (exploiting "energy" arguments), and a permanent construction (exploiting the Heilmann-Lieb theorem on matching polynomials). We conclude with a list of open questions.Comment: LaTeX2e, 111 pages. Submission includes Mathematica programs niceprincipal.m and nicetransversal.m Version 2 corrects a small error at the beginning of Appendix B, and makes a few small improvements elsewhere. To appear in Advances in Applied Mathematic

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 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 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. † Research supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392. Proposition 3.4. Let M be a matroid with ground set E, and let I, J be disjoint subsets of E. If M is Rayleigh and y > 0 then M I M J M IJ M. Proof. The inequality is trivial if either I or J is dependent, so assume that both I and J are independent in M. We first prove the result for I = {e 1 } and J = {f 1 , . . . , f k }. Notice that the Rayleigh difference of {e, f} in M may also be expressed as ∆M{e, f} = M e M f − M ef M. Thus, the Rayleigh condition is that if y > 0 then M e M f M ef M. Since every (contraction) minor

A polynomial P (x) in n complex variables is said to have the half-plane property if P (x) = 0 whenever all the variables have positive real parts. The generating polynomial for the set of all spanning trees of a graph G is one example. Motivated by the fact that the edge set of each spanning tree of G is a basis of the graphic matroid induced by G, it is shown by Choe et al. (Adv. Appl. Math. 32 (2004) 88-187) that the support of any homogeneous multiaffine polynomial with the half-plane property constitutes the set of all bases of a matroid. In this paper we show, when all the terms of a polynomial with the half-plane property have degrees of same parity, the support constitutes a jump system which is a generalization of matroids. Open problems and a few directions for further research will also be discussed.

Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219-235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.

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