Graph Embeddings, Symmetric Real Matrices, and Generalized Inverses

Graph embedding techniques for bounding eigenvalues of associated matrices have a wide range of applications. The bounds produced by these techniques are not in general tight, however, and may be off by a log n factor for some graphs. Guattery and Miller showed that, by adding edge directions to the graph representation, they could construct an embedding called the current flow embedding, which embeds each edge of the guest graph as an electric current flow in the host graph. They also showed how this embedding can be used to construct matrices whose nonzero eigenvalues had a one-to-one correspondence to the reciprocals of the eigenvalues of the generalized Laplacians. For the Laplacians of graphs with zero Dirichlet boundary conditions, they showed that the current flow embedding could be used generate the inverse of the matrix. In this paper, we generalize the definition of graph embeddings to cover all symmetric matrices, and we show a way of computing a generalized current flow embedding. We prove that, for any symmetric matrix A, the generalized current flow embedding of the orthogonal projector for the column space of A into A can be used to construct the generalized inverse, or pseudoinverse, of A. We also show how these results can be extended to cover Hermitian matrices.