Eigenvectors of graph Laplacians: a landscape

TL;DR

This paper reviews eigenvector properties of graph Laplacians, focusing on predicting eigenvalues from graph geometry and extending results on transformations that preserve or shift eigenvalues, enabling combinatorial eigenvalue analysis.

Contribution

It extends classical results on eigenvector properties, introduces combinatorial methods for eigenvalue prediction, and explores transformations linking graphs with specific eigenvalues.

Findings

Graphs with eigenvalues 1 to 6 can be generated from a short list of base graphs.

Transformations can predict eigenvalues from graph structure without numerical computation.

Conjecture that subgraphs and supergraphs with the same eigenvalue are connected by simple transformations.

Abstract

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on eigenvectors that have zero components and extend the pioneering results of Merris (1998) on graph transformations that preserve a given eigenvalue $\lambda$ or shift it in a simple way. These transformations enable us to obtain eigenvalues/vectors combinatorially instead of numerically; in particular we show that graphs having eigenvalues $\lambda= 1,2,\dots,6$ up to six vertices can be obtained from a short list of graphs. For the converse problem of a $\lambda$ subgraph $G$ of a $\lambda$ graph $G"$, we prove results and conjecture that $G$ and $G"$ are connected by two of the simple transformations described above.