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Infinite graphs with nonconstant Dirichlet finite harmonic functions. (English) Zbl 0752.31005
The paper exhibits a class of infinite graphs, each of which carries nonconstant harmonic functions with finite Dirichlet sum. (The graphs are embedded in a uniform way in the open unit circle with the hyperbolic metric, they have at least two accumulation points on the unit circle and they satisfy a strong isoperimetric inequality.) A method of constructing such harmonic functions with given boundary values is provided.

31C20 Discrete potential theory
05C10 Planar graphs; geometric and topological aspects of graph theory
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
51M20 Polyhedra and polytopes; regular figures, division of spaces
94C15 Applications of graph theory to circuits and networks
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