Coloquio de Matemáticas Aplicadas
We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. 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 subgraph $G$ of a graph $G”$, both affording $\lambda$, we prove results and conjecture that $G$ and $G”$ are connected by two of the simple transformations described above.

Imparte
Prof. Jean-Guy Caputo
Engineering Mathematics Department,
INSA de Rouen Technical University