Cycle-balance conditions for distance-regular graphs
It is known that in a distance-regular graph, the partition with respect to the 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. In this paper, the members of three families of distance-regular graphs, viz. the John graphs, Hamming graphs, and complete multipartite graphs have been considered. For each, all such sets which support an eigenvector for the next to the largest eigenvalue have been determined. These sets exhibit the underlying geometric structure of the graph.
scientific article; zbMATH DE number 1896029