  We discuss a particular kind of quantum walk on a general graph. We affix two
semi-infinite lines to a general finite graph, which we call tails. On the
tails, the particle making the walk simply advances one unit at each time step,
so that its behavior there is analogous to free propagation We are interested
in how many steps it will take the particle, starting on one tail and
propagating through the graph (where its propagation is not free), to emerge
onto the other tail. The probability to make such a walk in n steps and the
hitting time for such a walk can be expressed in terms of the transmission
amplitude for the graph, which is one element of its S matrix. Demonstrating
this necessitates a study of the analyticity properties of the transmission and
reflection amplitudes of a graph. We show that the graph can have bound states
that cannot be accessed by a particle entering the graph from one of its tails.
Time-reversal invariance of a quantum walk is defined and used to show that the
transmission amplitudes for the particle entering the graph from different
directions are the same if the walk is time-reversal invariant.
