On Uniformly Most Reliable Two-Terminal Graphs

A two‐terminal graph is an undirected graph G with vertex set V(G), edge set E(G), and two specified target vertices in V(G). If each edge of such a graph operates independently with the same fixed probability p, the two‐terminal reliability is the probability that there exists a path between the target vertices. A two‐terminal graph is uniformly most reliable if its reliability polynomial is greater than or equal to that of all other two‐terminal graphs with the same fixed number of vertices, n, and edges, m. In this article, we present specific values of n and m for which no uniformly most reliable two‐terminal simple graph exists, as well as values of n and m for which there does exist a uniformly most reliable two‐terminal simple graph.