A matrix S=(sij)∈Rn×n is said to determine a
\emph{transitional measure} for a digraph G on n vertices if for all
i,j,k∈{1,.˙.,n}, the \emph{transition inequality} sijsjk≤siksjj holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in G from i to k contains
j. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
d(⋅,⋅) is \emph{graph-geodetic}, that is, d(i,j)+d(j,k)=d(i,k)
holds if and only if every path in G connecting i and k contains j.
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic