  In previous papers about searches on star graphs several patterns have been
made apparent; the speed up only occurs when graphs are "tuned" so that their
time step operators have degenerate eigenvalues, and only certain initial
states are effective. More than that, the searches are never faster than order
root N time. In this paper the problem is defined rigorously, the causes for
all of these patterns are identified, sufficient and necessary conditions for
quadratic-speed searches for any connected subgraph are demonstrated, the
tolerance of these conditions is investigated, and it is shown that
(unfortunately) we can do no better than order root N time. Along the way, a
useful formalism is established that may be useful in future work involving
highly symmetric graphs.
