  Quantum walks are roughly analogous to classical random walks, and like
classical walks they have been used to find new (quantum) algorithms. When
studying the behavior of large graphs or combinations of graphs it is useful to
find the response of a subgraph to signals of different frequencies. In so
doing we can replace an entire subgraph with a single vertex with frequency
dependent scattering coefficients.
  In this paper a simple technique for quickly finding the scattering
coefficients of any quantum graph will be presented. These scattering
coefficients can be expressed entirely in terms of the characteristic
polynomial of the graph's time step operator. Moreover, with these in hand we
can easily derive the "impulse response" which is the key to predicting the
response of a graph to any signal. This gives us a powerful set of tools for
rapidly understanding the behavior of graphs or for reducing a large graph into
its constituent subgraphs regardless of how they are connected.
