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. K E Y W O R D Sgraph polynomial, optimal network, reliability polynomial, two-terminal reliability, two-terminal graph, uniformly most reliable Networks. 2018;00:1-17.wileyonlinelibrary.com/journal/net
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