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Gibbs random fields corresponding to systems of real-valued spins (e.g.systems of interacting anharmonic oscillators) indexed by the vertices ofunbounded degree graphs with a certain summability property are constructed. Itis proven that the set of tempered Gibbs random fields is non-void and weaklycompact, and that they obey uniform exponential integrability estimates. In thesecond part of the paper, a class of graphs is described in which the mentionedsummability is obtained as a consequence of a property, by virtue of whichvertices of large degree are located at large distances from each other. Thelatter is a stronger version of a metric property, introduced in [Bassalygo, L.A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with arandom potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf31} 572--589].

Gibbs random fields with unbounded spins on unbounded degree graphs