  A randomly walking quantum particle evolving by Schr\"odinger's equation
searches on $d$-dimensional cubic lattices in $O(\sqrt{N})$ time when $d \ge
5$, and with progressively slower runtime as $d$ decreases. This suggests that
graph connectivity (including vertex, edge, algebraic, and normalized algebraic
connectivities) is an indicator of fast quantum search, a belief supported by
fast quantum search on complete graphs, strongly regular graphs, and
hypercubes, all of which are highly connected. In this paper, we show this
intuition to be false by giving two examples of graphs for which the opposite
holds true: one with low connectivity but fast search, and one with high
connectivity but slow search. The second example is a novel two-stage quantum
walk algorithm in which the walking rate must be adjusted to yield high search
probability.
