Finding a Uniformly Most Reliable Graph

The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability 1 − p of each edge failing. A graph G is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of vertices and edges for all p ∈ [0, 1]. We examine the set of graphs with 8 vertices and 21 edges to determine if there exists a most reliable graph for this case. Specifically, we show that the graph whose compliment is 2K3 ∪ P2 is the uniformly most reliable graph with 8 vertices and 21 edges.