A New Family of Graph Representation Matrices: Application to Graph and Signal Classification

Most natural matrices that incorporate information about a graph are the adjacency and the Laplacian matrices. These algebraic representations govern the fundamental concepts and tools in graph signal processing even though they reveal information in different ways. Furthermore, in the context of spectral graph classification, the problem of cospectrality may arise and it is not well handled by these matrices. Thus, the question of finding the best graph representation matrix still stands. In this letter, a new family of representations that well captures information about graphs and also allows to find the standard representation matrices, is introduced. This family of unified matrices well captures the graph information and extends the recent works of the literature. Two properties are proven, namely its positive semidefiniteness and the monotonicity of their eigenvalues. Reported experimental results of spectral graph classification highlight the potential and the added value of this new family of matrices, and evidence that the best representation depends upon the structure of the underlying graph.