A topological interpretation of the walk distances

TL;DR

This paper offers a topological interpretation of walk distances in graphs by relating matrix cofactors and determinants through logarithmic identities, extending previous approaches to provide new insights.

Contribution

It introduces a novel topological interpretation of walk distances using matrix cofactors and logarithmic identities, extending classical determinant expansion methods.

Findings

Provides a new topological perspective on walk distances

Extends classical determinant expansion to cofactors of I-tA

Connects walk distances with matrix logarithm series

Abstract

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.