Assume that the vertices of a graph G are always operational, but the edges of G are operational independently with probability p ∈ [0, 1]. For fixed vertices s and t, the two‐terminal reliability of G is the probability that the operational subgraph contains an (s, t)‐path, while the all‐terminal reliability of G is the probability that the operational subgraph contains a spanning tree. Both reliabilities are polynomials in p, and have very similar behavior in many respects. However, unlike all‐terminal reliability polynomials, little is known about the roots of two‐terminal reliability polynomials. In a variety of ways, we shall show that the nature and location of the roots of two‐terminal reliability polynomials have significantly different properties than those held by roots of the all‐terminal reliability polynomials.

Assume that the vertices of a graph G are always operational, but the edges of G fail independently with probability q ∈ [0, 1]. The all-terminal reliability of G is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in q, and it was conjectured [5] that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when, for small simplicial complexes and matroids, the roots fall inside the closed unit disk.

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Given a connected graph G whose vertices are perfectly reliable and whose edges each fail independently with probability q ∈ [0, 1], the (all-terminal) reliability of G is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in q). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem-among all reliability polynomials of graphs on n vertices, what can we say about the rational roots? We prove that (for n ≥ 2), the rational roots are −1, − 1/2, − 1/3,…, − 1/(n − 1), 1. Moreover, we show that for n ≥ 3, the root of minimum modulus among all graphs of order n is rational, and determine all roots of smallest moduli and the corresponding graphs. Finally, we provide the first nontrivial mathematical property that distinguishes, via reliability, the class of simple graphs (i.e., those without loops and multiple edges) from that of graphs in general.