On graph Laplacian eigenvectors with components in {−1,0,1}

We characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {−1,+1} or {−1,0,+1}. Graphs having eigenvectors with components in {−1,+1} are called bivalent and are shown to be the regular bipartite graphs and their extensions obtained by adding edges between vertices with the same value for the given eigenvector. Graphs with eigenvectors with components in {−1,0,+1} are called trivalent and are shown to be soft-regular graphs – graphs such that vertices associated with non-zero components have the same degree – and their extensions via certain transformations.