Zeros of Reliability Polynomials and f-vectors of Matroids

Abstract

For a finite 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 ∈ [Copf ] is such that R_G(q) = 0 then [mid ]q[mid ] [les ] 1. We verify that this conjectured property of R_G(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.