By studying the voltage-current relations for electrical flows in a network, we found that both the Laplacian matrix and its pseudoinverse play an interesting role [1]. Perhaps more fascinating is that the orthogonal matrix of the eigenvectors of the Laplacian (and of its pseudoinverse) gives rise to a geometric representation of the corresponding graph. Thus, a graph with N nodes and L links can be represented and studied equivalently (i.e. without losing information) in three domains:
(a) the topological domain, represented by an N x N adjacency matrix or node/link lists, is by far the most familiar study environment of graphs
(b) the spectral domain represents the eigenstructure (eigenvalues and eigenvectors) of a graph related matrix (such as the adjacency and Laplacian) matrix and
(c) an N-1 dimensional, Euclidean space, in which each graph can be represented by a simplex with N vertices; each vertex corresponds to a node.
In this talk, we will mainly focus on domain (c) and we will explore geometric properties of graphs.
References:[1] Van Mieghem, P., K. Devriendt and H. Cetinay, 2017, "Pseudoinverse of the Laplacian and best spreader node in a network", Physical Review E, vol. 96, No. 3, p 032311.