  A new class of distances for graph vertices is proposed. This class contains
parametric families of distances which reduce to the shortest-path, weighted
shortest-path, and the resistance distances at the limiting values of the
family parameters. The main property of the class is that all distances it
comprises are graph-geodetic: $d(i,j)+d(j,k)=d(i,k)$ if and only if every path
from $i$ to $k$ passes through $j$. The construction of the class is based on
the matrix forest theorem and the transition inequality.
