A class of graph-geodetic distances generalizing the shortest-path and the resistance distances

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)d(i,j)+d(j,k)=d(i,k) if and only if every path from ii to kk passes through jj. The construction of the class is based on the matrix forest theorem and the transition inequality.