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We prove various results connecting structural or algebraic properties ofgraphs and groups to conditions on their spaces of harmonic functions. Inparticular: we show that a group with a finitely supported symmetric measurehas a finite-dimensional space of harmonic functions if and only if it isvirtually cyclic; we present a new proof of a result of V. Trofimov that aninfinite vertex-transitive graph admits a non-constant harmonic function; wegive a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J.Dodziuk that the Laplacian on an infinite, connected, locally finite graph issurjective; and we show that the positive harmonic functions on a non-virtuallynilpotent linear group span an infinite-dimensional space.

Characterisations of algebraic properties of groups in terms of harmonic functions