# Gomory-Hu trees of infinite graphs with finite total weight

## TL;DR

This paper extends the concept of Gomory-Hu trees from finite graphs to infinite weighted graphs with finite total weight, demonstrating the existence of such trees in this broader context.

## Contribution

It generalizes Gomory-Hu trees to infinite graphs with finite total weight, a significant extension of the classical finite case.

## Key findings

- Gomory-Hu trees exist for infinite graphs with finite total weight.
- An example shows the necessity of the finite total weight condition.
- The result broadens the applicability of Gomory-Hu trees to infinite graph settings.

## Abstract

Gomory and Hu proved that if $ G $ is a finite graph with non-negative weights on its edges, then there exists a tree $ T $ (called now Gomory-Hu tree) on $ V(G) $ such that for all $ u\neq v\in V(G) $ there is an $ e\in E(T) $ such that the two components of $ T-e $ determines an optimal (minimal valued) cut between $ u $ an $ v $ in $ G $. In this paper we extend their result to infinite weighted graphs with finite total weight. Furthermore, we show by an example that one can not omit the condition of finiteness of the total weight.