The walk distances in graphs

The walk distances in graphs are defined as the result of appropriate transformations of the ∑k=0∞(tA)k proximity measures, where AA is the weighted adjacency matrix of a graph and tt is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter tt approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a specific subclass of walk distances. On the other hand, the long walk distance is equal to the resistance distance in a transformed graph.