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Many approaches to image classification tend to transform an image into an unstructured set of numeric feature vectors obtained globally and/or locally, and as a result lose important relational information between regions. In order to encode the geometric relationships between image regions, we propose a variety of structural image representations that are not specialised for any particular image category. Besides the traditional grid-partitioning and global segmentation methods, we investigate the use of local scale-invariant region detectors. Regions are connected based not only upon nearest-neighbour heuristics, but also upon minimum spanning trees and Delaunay triangulation. In order to maintain the topological and spatial relationships between regions, and also to effectively process undirected connections represented as graphs, we utilise the recently-proposed graph neural network model. To the best of our knowledge, this is the first utilisation of the model to process graph structures based on local-sampling techniques, for the task of image classification. Our experimental results demonstrate great potential for further work in this domain.