Eigenvalues of Transmission Graph Laplacians

The standard notion of the Laplacian of a graph is generalized to the setting of a graph with the extra structure of a “transmission” system. A transmission system is a mathematical representation of a means of transmitting (multi-parameter) data along directed edges from vertex to vertex. The associated transmission graph Laplacian is shown to have many of the former properties of the classical case, including: an upper Cheeger type bound on the second eigenvalue minus the first of a geometric isoperimetric character, relations of this difference of eigenvalues to diameters for k-regular graphs, eigenvalues for Cayley graphs with transmission systems. An especially natural transmission system arises in the context of a graph endowed with an association. Other relations to transmission systems arising naturally in quantum mechanics, where the transmission matrices are scattering matrices, are made. As a natural merging of graph theory and matrix theory, there are numerous potential applications, for example to random graphs and random matrices. §1: Introduction. §2: Definition of the Transmission Graph Laplacian and the Generalized Cheeger Estimate. §3: A Symmetric Graph Transmission System for Associations. §4: Quantum Mechanical Example of Transmission Systems: Invertible and (Hermitian) Symmetric. §5: The Dual of a Graph and its Transmission Systems. §6: Estimates 1: Range of Values and Cheeger’s Upper Bound Generalized. §7: Estimates 2: Eigenvalues and Diameters. §8: Reinterpretation in terms of Capacities. Cheeger Constants. §9: Cayley graphs. Their Eigenvalues. §10: Eigenvalues for Graph Collapses and Amalgamations. §11: A Categorical Approach. The push forward collapse. §12: Morse Theory, Riemann Surfaces and Transmission systems.