Spectral Analysis of Complex Laplacian Matrices
This paper explores how to extend the spectral analysis of graphs to the case where the nodes and edges are attributed. To do this we introduce a complex Hermitian variant of the Laplacian matrix. Our spectral representation is based on the eigendecomposition of the resulting Hermitian property matrix. The eigenvalues of the matrix are real while the eigenvectors are complex. We show how to use symmetric polynomials to construct permutation invariants from the elements of the resulting complex spectral matrix. We construct pattern vectors from the resulting invariants, and use them to embed the graphs in a low dimensional pattern space using a number of well-known techniques including principal components analysis, linear discriminant analysis and multidimensional scaling.