A topological interpretation of the walk distances

The walk distances in graphs have no direct interpretation in terms of walk weights, since they are introduced via the \emph{logarithms} of walk weights. Only in the limiting cases where the logarithms vanish such representations follow straightforwardly. The interpretation proposed in this paper rests on the identity \ln\det B=\tr\ln B applied to the cofactors of the matrix $I-tA,$ where $A$ is the weighted adjacency matrix of a weighted multigraph and $t$ is a sufficiently small positive parameter. In addition, this interpretation is based on the power series expansion of the logarithm of a matrix. Kasteleyn (1967) was probably the first to apply the foregoing approach to expanding the determinant of $I-A$. We show that using a certain linear transformation the same approach can be extended to the cofactors of $I-tA,$ which provides a topological interpretation of the walk distances.Comment: 13 pages, 1 figure. Version #

Simple expressions for the long walk distance

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 connected weighted 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. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix ${\cal L}=\rho I-A,$ where $\rho$ is the Perron root of $A.$Comment: 7 pages. Accepted for publication in Linear Algebra and Its Application

The Walk Distances in Graphs

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.Comment: Accepted for publication in Discrete Applied Mathematics. 26 pages, 3 figure