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

Effective and efficient approximations of the generalized inverse of the  graph Laplacian matrix with an application to current-flow betweenness  centrality