Abstract : This grant has supported work in several areas. 1) A study of graph eigenvectors shows connections to graph structure in ways that are reminiscent of eigenfunctions of the laplacian operator in two or three dimensions. Methods developed in this study have also led to estimates of the maximum possible value for the kth eigenvalue of a graph as function of the number of edges or vertices. 2) The convex hull of the rows of an eigenmatrix of a graph is the polytope of an eigenvalue. We investigated relations between such polytopes and the graph. The graph of such a polytope may be isomorphic to the original graph this is the case for most regular polytopes. For distance-regular graphs and several kinds of less symmetric graphs, we can show that the polytope of some eigenvalue has the same group of automorphisms as the graph, that proximity of points is equivalent to adjacency of vertices, and that other properties of the polytope carry over the graph. Possible directions for future work include the following. Determine the reducibility of the group of automorphisms of a polytope and the significance in the graph of faces and facets of the polytopes. Investigate the intrinsic eigenvectors of a graph (the list of inner products of vertices of a polytope with the normal to a supporting hyperplane is an intrinsic eigenvector). Seek physical models for the interpretation of graph eigenvalues and eigenvectors, e.g., transient temperature distributions in a graph-like collection of heat-conducting rods.