The Walk Distances in Graphs

Details The Walk Distances in Graphs The Walk Distances in Graphs

2011-03-10

arxiv.org/abs/1103.2059v8

Mathematics

Combinatorics

Accepted for publication in Discrete Applied Mathematics. 26 pages, 3 figures

Scientific paper

10.1016/j.dam.2012.02.015

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.

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