  We devise methods for finding approximations of the generalized inverse of
the graph Laplacian matrix, which arises in many graph-theoretic applications.
Finding this matrix in its entirety involves solving a matrix inversion
problem, which is resource demanding in terms of consumed time and memory and
hence impractical whenever the graph is relatively large. Our approximations
use only few eigenpairs of the Laplacian matrix and are parametric with respect
to this number, so that the user can compromise between effectiveness and
efficiency of the approximated solution. We apply the devised approximations to
the problem of computing current-flow betweenness centrality on a graph.
However, given the generality of the Laplacian matrix, many other applications
can be sought. We experimentally demonstrate that the approximations are
effective already with a constant number of eigenpairs. These few eigenpairs
can be stored with a linear amount of memory in the number of nodes of the
graph and, in the realistic case of sparse networks, they can be efficiently
computed using one of the many methods for retrieving few eigenpairs of sparse
matrices that abound in the literature.
