The relevance of quenched, uncorrelated disorder coupling to the local energy density, its paradigm being the random-bond model, is judged by the Harris criterion. A generalization of the underlying argument to the case of spatially correlated disorder, exemplified by quasi-crystals, has been given by Luck. We address the question, whether a relevance criterion of this type is applicable to the case of spin models coupled to different kinds of  random graphs. The geometrical fluctuation exponent appearing in Luck's criterion is precisely determined for the cases of two-dimensional Poissonian Voronoie--Delaunay random lattices and planar, ``fat'' \phi ³ Feynman diagrams. While previous work for the latter graphs is in accord with the determined relevance threshold, a preliminary analysis of the results of a Monte Carlo simulation of the three-states Potts model on Poissonian Voronoie\ lattices presented here does not meet the expectations from the relevance criterion.

PACS numbers: 75.10.Hk, 75.40.Mg, 75.50.Lk