Harmonic functions on annuli of graphs (Q699485)

The author proves the ``relative connectedness'' of graphs which satisfy a polynomial volume growth and a Poincaré-type inequality on balls. By ``relative connectedness'' it is meant that every two vertices at distance \(R\) from a vertex \(x\) can be joined by a path within an annulus. In the case of Cayley graph of groups having polynomial volume growth, the above result uses to obtain a Poincaré-type inequality on the annuli.