Abstract: The walk distances in graphs are defined as the result of appropriate transformations of the P1 k=0(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. Furthermore, 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.

Keywords: graph distances, walk distances, logarithmic forest distances, transitional measure, Laplacian matrix, resistance distance, network

ACM Classification Keywords: G.2.2 Graph Theory – Network problems; E.1 Data Structures – Graphs and networks; C.2.1 Network Architecture and Design – Network topology

Link:

LOGARITHMIC DISTANCES IN GRAPHS

Pavel Chebotarev

http://www.foibg.com/ibs_isc/ibs-25/ibs-25-p03.pdf