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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.
##### MSC:
 31C20 Discrete potential theory and numerical methods 05C40 Connectivity
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##### References:
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