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In this paper, we describe the use of concepts from the areas of structural and statistical pattern recognition for the purposes of recovering a mapping which can be viewed as an operator on the graph attribute-set. This mapping can be used to embed graphs into spaces where tasks such as classification and retrieval can be effected. To do this, we depart from concepts in graph theory so as to introduce mappings as operators over graph spaces. This treatment leads to the recovery of a mapping based upon the graph attributes which is related to the edge-space of the graphs under study. As a result, the recovered mapping is a linear operator over the attribute set which is associated with the graph topology. To recover this mapping, we employ an optimisation approach whose cost function is based upon the Chi-squared distance and is related to the target function used in discrete Markov Random Field approaches. Thus, the method presented here provides a link between concepts in graph theory, statistical inference and linear operators. We illustrate the utility of the recovered embedding for purposes of shape categorisation and retrieval. We also compare our results to those yielded by alternatives.