In [7], several conjectures are listed
about uniformly most reliable graphs, and, to date, no
counterexamples have been found. These include the conjectures
that an optimal reliable graph has degrees that are almost
regular, has maximum girth, and has minimum diameter. In
this article, we consider simple graphs and present one
counterexample and another possible counterexample of these
conjectures: maximum girth (i.e., maximize the length of
the shortest circuit of the graph G) and minimum
diameter (i.e., minimize the maximum possible distance
between any pair of vertices).