Spectral Analysis of Complex Laplacian Matrices

Abstract

Abstract. 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.

A New Surrogating Algorithm by the Complex Graph Fourier Transform (CGFT)

“…In [ 21 ] a Hermitian Laplacian matrix is proposed to include some possible oriented edges, but this leads to a very limited form of Hermitian Laplacian matrices—Its off-diagonal elements are 1 for non-oriented edges and for the oriented edge. A more general Hermitian Laplacian matrix is proposed in [ 22 ] by adding a phase term to the off-diagonal elements of a real Laplacian matrix. Then some permutation invariants are found in the pattern recognition context.…”

Synthesis of Median Spectral Graph

“…The method was restricted to AGs with only one positive attribute on the nodes and arcs. Recently, AGs with complex numbers as attributes on the nodes or arcs were allowed in the method presented in [9,10], rather than purely real entries.…”