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Although graph structures have proved useful in high level vision for object recognition and matching, they can prove computationally cumbersome because of the need to establish reliable correspondences between nodes. Hence, standard pattern recognition techniques cannot be easily applied to graphs since feature vectors are not easily constructed. To overcome this problem, we turn to the spectral matrix. We show how the elements of this matrix can be used to construct symmetric polynomials that are permutation invariant. The coefficients of these polynomials can be used as graph-features which can be encoded in a vectorial manner. Hence, the symmetric polynomials lead to a representation which is invariant under node permutations and so represents the graph structure without the need for labelling or correspondence operations. We demonstrate that these features are complete and continuous for 'simple' graphs (those without repeated eigenvalues in their spectrum). The notions of stability and discrimination are discussed, and we present experimental evaluation of these properties. Finally, we show that these graph characterizations can be used to cluster graphs from real datasets.