  We characterize the spectrum of the transition matrix for simple random walk
on graphs consisting of a finite graph with a finite number of infinite Cayley
trees attached. We show that there is a continuous spectrum identical to that
for a Cayley tree and, in general, a non-empty pure point spectrum. We apply
our results to studying continuous time quantum walk on these graphs. If the
pure point spectrum is nonempty the walk is in general confined with a nonzero
probability.
