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The inverse geodesic length of a graphG is the sum of the inverse of the distances between all pairs of distinct vertices ofG . In some domains, it is known as the Harary index or the global efficiency of the graph. We show that, ifG is planar and hasn vertices, then the inverse geodesic length ofG can be computed in roughlyO (n 9/5 ) time. We also show that, ifG hasn vertices and treewidth at mostk , then the inverse geodesic length ofG can be computed inO (n logO (k )n ) time. In both cases, we use techniques developed for computing the sum of the distances, which does not have “inverse” component, together with batched evaluations of rational functions.

Computing the Inverse Geodesic Length in Planar Graphs and Graphs of Bounded Treewidth