# Distance-balanced graphs and Travelling Salesman Problems

## TL;DR

This paper introduces a new probabilistic graph property called pTS-distance-balancedness, linking it to classical concepts and the Traveling Salesman Problem, and characterizes it through graph compositions like wreath products.

## Contribution

It defines pTS-distance-balancedness, relates it to probabilistic centrality measures, and characterizes this property for wreath product graphs, connecting it to classical distance-balancedness.

## Key findings

- pTS-distance-balancedness generalizes classical distance-balancedness.
- Characterization of pTS-distance-balanced graphs via probabilistic centrality.
- Wreath product graphs' distance-balancedness depends on factors' properties.

## Abstract

For every probability $p\in[0,1]$ we define a distance-based graph property, the $p$TS-distance-balancedness, that in the case $p=0$ coincides with the standard distance-balancedness, and in the case $p=1$ is related to the Hamiltonian-connectedness. In analogy with the classical case, where the distance-balancedness of a graph is equivalent to the property of being self-median, we characterize the class of $p$TS-distance-balanced graphs in terms of their equity with respect to certain probabilistic centrality measures, inspired by the Travelling Salesman Problem. We prove that it is possible to detect this property looking at the classical distance-balancedness (and therefore looking at the classical centrality problems) of a suitable graph composition, namely the wreath product of graphs. More precisely, we characterize the distance-balancedness of a wreath product of two graphs in terms of the $p$TS-distance-balancedness of the factors.