Graph Embeddings and Laplacian Eigenvalues

• Authors: Stephen Guattery;Gary L. Miller
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2000

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n × n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix $\Gamma$; the best possible bound based on this embedding is $n/\lambda_{\max} (\Gamma^T \Gamma)$, where $\lambda_{\max}$ indicates the largest eigenvalue of the specified matrix. However, the best bounds produced by embedding techniques are not tight; they can be off by a factor proportional to log2 n for some Laplacians.We show that this gap is a result of the representation of the embedding: By including edge directions in the embedding matrix representation $\Gamma$, it is possible to find an embedding such that $\Gamma^T \Gamma$ has eigenvalues that can be put into a one-to-one correspondence with the eigenvalues of the Laplacian. Specifically, if $\lambda$ is a nonzero eigenvalue of either matrix, then $n / \lambda$ is an eigenvalue of the other. 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. In the zero Dirichlet boundary case, the embedding matrix $\Gamma$ can be used to construct the inverse of the Laplacian. Finally, we connect our results with previous techniques for producing bounds and provide an example.