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

TL;DR

This paper characterizes specific classes of graphs based on the structure of their Laplacian eigenvectors with components in {-1,0,1}, revealing their relation to regular bipartite and soft-regular graphs.

Contribution

It provides a complete characterization of graphs with Laplacian eigenvectors having components in {-1,0,1}, identifying bivalent and trivalent graph classes and their extensions.

Findings

Bivalent graphs are regular bipartite and extendable by same-value edge additions.

Trivalent graphs are soft-regular with vertices of non-zero components sharing the same degree.

Extensions of these graphs are characterized by specific transformations.

Abstract

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 some transformations.