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.

💡 Research Summary

The paper introduces a substantial generalization of the classical graph Laplacian by incorporating a “transmission system” that assigns an (m\times m) complex matrix to each directed edge. This matrix models the linear transformation of multi‑parameter data (such as colors, spins, voltages, or quantum amplitudes) as it travels from one vertex to another. The resulting operator, called the transmission graph Laplacian (\Delta_T), is defined for a function (f:V\to\mathbb{C}^m) by
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