Graph Embeddings and Laplacian Eigenvalues SIAM Journal on Matrix Analysis and Applications Volume 21, Number 3 pp. Gary L. Miller Abstract. Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. The lower bound is based on a clique embedding into the underlying graph of the Laplacian.
Simple transformations map the corresponding eigenvectors to each other.
The embedding that produces these correspondences has a simple description in electrical terms if the underlying graph of the Laplacian is viewed as a resistive circuit. We also show that a similar technique works for star embeddings when the Laplacian has a zero Dirichlet boundary condition, though the related eigenvalues in this case are reciprocals of each other. Finally, we connect our results with previous techniques for producing bounds and provide an example. Laplacian matrices, graph eigenvalues and eigenvectors, graph embeddings AMS Subject Classifications . 05C50, 15A18, 15A42 PII .