โ Scribed by Liu, Shaobin; Cheng, Kam-Hoi; Liu, Xiaoping
No coin nor oath required. For personal study only.
Given a graph whose edges never fail but whose nodes fail independently of each other with a constant probability 1 ---p p p, the reliability of a graph is defined to be the probability that the induced subgraph of the surviving nodes is connected. Let โฆ (n n n, m m m) be the class of all graphs with n n n nodes and m m m edges. A graph G of โฆ is said to be uniformly best in โฆ, if for all choices of p p p, it is most reliable in the class of graphs, โฆ. In this paper, the existing known set of uniformly best graphs is extended, and some closely related graphs are proved to be not the uniformly best graphs in their class. More precisely, we prove that for any positive integer b b b the complete tripartite graph K K K(b b b, b b b + + + 1, b b b + + + 2) is uniformly best in its class โฆ(3b b b + + + 3, 3b b b 2 + + + 6b b b + + + 2), while the complete tripartite graphs K K K(b b b, b b b + + + 1, b b b +
are not the uniformly best in their classes โฆ(3b b b + + + i i i + + + 1, 3b b b 2 + + + 2(i i i + + + 1)b b b + + + i i i).
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