## 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 05C40 Connectivity
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### References:

 [1] Coulhon, T. and Saloff-Coste, L., Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana9 (1993), no. 2, 293-314. · Zbl 0782.53066 [2] Delmotte, T., Inégalité de Harnack elliptique sur les graphes, Colloquium Mathematicum80 (1997), no. 1, 19-37. · Zbl 0871.31008 [3] Delmotte, T., Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana15 (1999), no. 1, 181-232. · Zbl 0922.60060 [4] Diaconis, P. and Saloff-Coste, L., Comparison theorems for reversible Markov chains, Ann. Appl. Probab. 3 (1993), no. 3, 696-730. · Zbl 0799.60058 [5] Hajłasz, P. and Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. · Zbl 0954.46022 [6] Moser, J., On Harnack’s theorem for elliptic differential equations, Comm. Pure and Applied Math. 14 (1961), 577-591. · Zbl 0111.09302 [7] Saloff-Coste, L., Isoperimetric inequalities and decay of iterated kernels for almost-transitive Markov chains, Combin. Probab. Comput. 4 (1995), 419-442. · Zbl 0842.60070
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