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: $d(i,j)+d(j,k)=d(i,k)$ 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.

💡 Research Summary

The paper introduces a novel family of graph‑vertex distances that unifies several classical metrics—namely the shortest‑path distance, the weighted shortest‑path distance, and the resistance distance—through a single parametric framework. The authors begin by recalling the standard definitions: the shortest‑path distance measures the minimum sum of edge weights along any path, while the resistance distance derives from the Moore‑Penrose pseudoinverse of the Laplacian and reflects effective electrical resistance between nodes. Both capture different aspects of network structure, yet they lack a common theoretical foundation.

To bridge this gap, the authors exploit the Matrix Forest Theorem, which states that for a graph with Laplacian (L) and a positive scalar (\alpha), the matrix \