Eigenvalues of Transmission Graph Laplacians

TL;DR

This paper generalizes the graph Laplacian to include transmission systems, exploring its properties and applications in areas like quantum mechanics, random graphs, and matrix theory.

Contribution

It introduces the transmission graph Laplacian, extending classical graph Laplacian properties to new settings with transmission systems.

Findings

Eigenvalues satisfy Cheeger-type bounds.

Relations between eigenvalues and graph diameters.

Applications to Cayley graphs and quantum mechanics.

Abstract

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.