The graph bottleneck identity

A matrix S = (sij) ∈ R n×n is said to determine a transitional measure for a digraph Γ on n vertices if for all i, j, k ∈ {1, . . . , n}, the transition inequality sij sjk ≤ sik sjj holds and reduces to the equality (called the graph bottleneck identity) if and only if every path in Γ 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 graph-geodetic, that is, d(i, j) + d(j, k) = d(i, k) holds if and only if every path in Γ 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.