Roots of the reliability polynomial
SIAM Journal on Discrete Mathematics
Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra
SIAM Journal on Discrete Mathematics
The Combinatorics of Network Reliability
The Combinatorics of Network Reliability
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
Combinatorics, Probability and Computing
Zeros of Reliability Polynomials and f-vectors of Matroids
Combinatorics, Probability and Computing
The Brown--Colbourn conjecture on zeros of reliability polynomials is false
Journal of Combinatorial Theory Series B
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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.