Eigenvectors of Graph-Laplace-Operators

The foundations of spectral graph theory were laid in the fifties and sixties. Since then, spectral methods have become standard techniques in (algebraic) graph theory.
The eigenvalues of graphs, most often defined as the eigenvalues of the adjacency matrix, have received much attention over the last thirty years as a means of characterizing classes of graphs and for obtaining bounds on properties such as the diameter, girth, chromatic number, connectivity, etc. More recently, the interest has shifted somewhat from the adjacency spectrum to the spectrum of the closely related graph Laplacian. The eigenvectors of graphs, however, have received only sporadic attention on their own. We investigate the geometric features of the eigenfunctions of discrete Laplacian operators depending on their location in the spectrum and depending on the structure of the underlying graph.

Austrian Science Fund