Pseudoinverses of Signed Laplacian Matrices

Abstract

Even for nonnegative graphs, the pseudoinverse of a Laplacian matrix is not an "ordinary" (i.e., unsigned) Laplacian matrix, but rather a signed Laplacian. In this paper, we show that the property of eventual positivity provides a natural embedding class for both signed and unsigned Laplacians, class which is closed with respect to pseudoinversion as well as to stability. Such class can deal with both undirected and directed graphs. In particular, for digraphs, when dealing with pseudoinverse-related quantities such as effective resistance, two possible solutions naturally emerge, differing in the order in which the operations of pseudoinversion and of symmetrization are performed. Both lead to an effective resistance which is a Euclidean metric on the graph.