Estimating noncommutative distances on graphs

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 $G$.

24 p, 4 figures. Comments welcome !

References in corpus (3)

• The distance formula in algebraic spacetime theories
• Gauge networks in noncommutative geometry
• Canonical quantization and the spectral action, a nice example