Blachère, Sébastien Harmonic functions on annuli of graphs. (English) Zbl 1005.31005 Ann. Math. Blaise Pascal 8, No. 2, 47-59 (2001). 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. Reviewer: Sophia L.Kalpazidou (Thessaloniki) MSC: 31C20 Discrete potential theory 05C40 Connectivity Keywords:relative connectedness of graphs; Poincaré-type inequality; Cayley graph of groups PDF BibTeX XML Cite \textit{S. Blachère}, Ann. Math. Blaise Pascal 8, No. 2, 47--59 (2001; Zbl 1005.31005) Full Text: DOI Numdam EuDML OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.