  In this paper, we obtain a lower bound for the smallest eigenvalue of a
regular graph containing many copies of a smaller fixed subgraph. This
generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a
triangle. We apply our results to obtain a lower bound on the smallest
eigenvalue of the associahedron graph, and we prove that this bound gives the
correct order of magnitude of this eigenvalue. We also survey what is known
regarding the second-largest eigenvalue of the associahedron graph.
