Cycle-balance conditions for distance-regular graphs

Abstract

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determine all such sets which support an eigenvector for the next to largest eigenvalue. These sets exhibit the underlying geometric structure of the graph.