Diffusion characters: Breaking the spectral barrier

Combinatorial graphs are used as mathematical models of a broad variety of phenomena including communications networks, gene regulation networks, food webs, or even to map out resource conflicts. The diffusion character matrix of a graph injects the vertices of a graph into Euclidean space so that Euclidean distances between vertices are closely tied to connectivity between those vertices in the graph. In this paper diffusion characters and their associated matrices are defined, elementary properties are derived, and it is demonstrated that diffusion character matrices contain information not contained in the eigenvalues of the graph. This latter property is demonstrated by computing the diffusion character matrices of two famous co-spectral graphs, two of the three (3,10)-cages, which are cubic graphs on 70 vertices.