  The walk distances in graphs are defined as the result of appropriate
transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$
is the weighted adjacency matrix of a graph and $t$ 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 $t$ 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 subclass of walk distances. On
the other hand, the long walk distance is equal to the resistance distance in a
transformed graph.
