Crossing properties of graph reliability functions

Let G be a graph and p ϵ (0, 1). Let A(G, p) denote the probability that if each edge of G is selected at random with probability p then the resulting spanning subgraph of G is connected. Then A(G, p) is a polynomial in p. We prove that for every integer k ≥ 1 and every k‐tuple (m₁, m₂, … ,m_k) of positive integers there exist infinitely many pairs of graphs G₁ and G₂ of the same size such that the polynomial A(G₁, p) − A(G₂, p) has exactly k roots x₁ < x₂ < ··· < x_k in (0, 1) such that the multiplicity of x_i is m_i. We also prove the same result for the two‐terminal reliability polynomial, defined as the probability that the random subgraph as above includes a path connecting two specified vertices. These results are based on so‐called A‐ and T‐multiplying constructions that are interesting in themselves. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 206–221, 2000