We study the walk distances for graph vertices, which are obtained by meansof appropriate transformations from the $\sum_{k=0}^\infty(tA)^k$ proximity measure, 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.
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