Estimating noncommutative distances on graphs

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 $G$.