Harmonic functions on annuli of graphs. (English) Zbl 1005.31005

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.


31C20 Discrete potential theory
05C40 Connectivity
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