In this paper, we describe the use of concepts from structural and statistical pattern recognition for 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 categorisation and relational matching can be effected. We depart from concepts in graph theory 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, this mapping is a linear operator over the attribute set which is associated with the graph topology. Here, we employ an optimisation approach whose cost function is related to the target function used in discrete Markov Random Field approaches. Thus, the proposed method provides a link between concepts in graph theory, statistical inference and linear operators. We illustrate the utility of the recovered embedding for shape matching and categorisation on MPEG7 CE-Shape-1 dataset. We also compare our results to those yielded by alternatives.