Eigenvectors of graph Laplacians: a landscape

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.