Roots of the reliability polynomial
SIAM Journal on Discrete Mathematics
Zeros of rank-generating functions of Cohen-Macaulay complexes
Proceedings of the 4th conference on Formal power series and algebraic combinatorics
Discrete Mathematics
Non-Stanley Bounds for Network Reliability
Journal of Algebraic Combinatorics: An International Journal
The Combinatorics of Network Reliability
The Combinatorics of Network Reliability
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
Combinatorics, Probability and Computing
The Brown--Colbourn conjecture on zeros of reliability polynomials is false
Journal of Combinatorial Theory Series B
Polynomials with the half-plane property and the support theorems
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series B
| Hi-index | 0.00 |
For a finite multigraph G, the reliability function of G is the probability RG(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 ∈ ℂ is such that RG(q) = 0 then ∣q∣ ≤ 1. We verify that this conjectured property of RG(q) holds if G is a series-parallel network. The proof is by an application of the Hermite–Biehler theorem and development of a theory of higher-order interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the f-vector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the Brown–Colbourn conjecture, and are hence true for cographic matroids of series-parallel networks.