Estimating noncommutative distances on graphs

TL;DR

This paper explores Connes' noncommutative distance on graphs, establishing a lower bound related to geodesic distance and graph degree using spectral triples, advancing understanding in noncommutative geometry on discrete structures.

Contribution

The paper introduces a new lower bound for noncommutative distance on graphs, connecting it to geodesic distance and graph degree via spectral triples.

Findings

Established a lower bound for noncommutative distance in graphs

Connected noncommutative distance to geodesic distance and graph degree

Utilized spectral triples on graph edges for analysis

Abstract

We report on some findings concerning Connes' noncommutative distance $d$ on a weighted undirected graph $G$. Our main result is the lower bound $\ell/\Delta(G)\le d$ where $\ell$ is the geodesic distance and $\Delta(G)$ the degree of $G$. It is obtained thanks to an auxiliary spectral triple on the collection of the edges of $G$.