A Class of Graph-Geodetic Distances Generalizing the Shortest-Path and the Resistance Distances

TL;DR

This paper introduces a new class of graph distances that unify shortest-path and resistance distances, ensuring all are graph-geodetic and based on matrix forest theory.

Contribution

It proposes a parametric family of graph distances that generalize key existing distances and are graph-geodetic, based on matrix forest theorem and transition inequality.

Findings

Includes shortest-path, weighted shortest-path, and resistance distances as special cases.

All distances in the class are graph-geodetic.

The class is constructed using matrix forest theorem and transition inequality.

Abstract

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.