  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 $A.$
