Graph pattern spaces from Laplacian spectral polynomials

Graph structures have proved computationally cumbersome for pattern analysis. The reason for this is that before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The co-efficients of these polynomials can be used as graph-features which can be encoded in a vectorial manner. We explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis (PCA), multidimensional scaling (MDS) and locality preserving projection (LPP).