Circle boundaries of planar graphs |
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Authors: | S Northshield |
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Institution: | (1) Department of Mathematics, SUNY, 12901 Plattsburgh, NY, USA |
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Abstract: | Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle. |
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Keywords: | Primary: 60J50 31C35 secondary 05C38 31C05 |
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