A new class of distances for graph vertices is proposed. This class containsparametric families of distances which reduce to the shortest-path, weightedshortest-path, and the resistance distances at the limiting values of thefamily parameters. The main property of the class is that all distances itcomprises are graph-geodetic: $d(i,j)+d(j,k)=d(i,k)$ if and only if every pathfrom $i$ to $k$ passes through $j$. The construction of the class is based onthe matrix forest theorem and the transition inequality.
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机译：提出了一类新的图顶点距离。此类包含距离的参数族，这些距离的族减少为最短路径，加权最短路径，以及在族参数的极限值处的电阻距离。该类的主要特性是，它包含的所有距离都是图大地坐标：$ d（i，j）+ d（j，k）= d（i，k）$当且仅当从$ i $到$ k的每条路径$通过$ j $。该类的构建基于矩阵森林定理和过渡不等式。
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