Counterexamples to conjectures for uniformly optimally reliable graphs

In [7], several conjectures are listed about uniformly most reliable graphs , and, to date, no counterexamples have been found. These include the conje ctures that an optimal reliable graph has degrees that are almost regular, has maximum girth, and has minimum diameter. In this article, we consider s imple graphs and present one counterexample and another possible counterexa mple 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 m aximum possible distance between any pair of vertices).