Simple expressions for the long walk distance

TL;DR

This paper derives simple mathematical expressions for the long walk distance in graphs, connecting it to matrix inverses and minors of a specific singular M-matrix, enhancing understanding of graph distances.

Contribution

It provides new, simplified formulas for the long walk distance using generalized inverse, minors, and submatrix inverses of a key M-matrix.

Findings

Derived explicit formulas for the long walk distance

Connected long walk distance to matrix inverses and minors

Enhanced computational understanding of graph distances

Abstract

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 $A.$