Abstract.
Following the Euclidean example, we introduce the strong and weak mean value property for finite variation measures on graphs. We completely characterize finite variation measures with bounded support on radial trees which have the strong mean value property. We show that for counting measures on bounded subsets of a tree with root o, the strong mean value property is equivalent to the invariance of the subset under the action of the stabilizer of o in the automorphism group. We finally characterize, using the discrete Laplacian, the finite variation measures on a generic graph which have the weak mean value property and we give a non-trivial example.
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Received: July 21, 2000; in final form: March 13, 2001¶Published online: March 19, 2002
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Zucca, F. The mean value property for harmonic functions on graphs and trees. Ann. Mat. Pura Appl. IV. Ser. 181, 105–130 (2002). https://doi.org/10.1007/s102310200032
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DOI: https://doi.org/10.1007/s102310200032