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arXiv · 0912.4048

Eigenvalues of Transmission Graph Laplacians

Sylvain E. Cappell·Edward Y. Miller
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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.

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Sylvain E. Cappell, Edward Y. Miller. 2009-12-20. Eigenvalues of Transmission Graph Laplacians. https://arxiv.org/abs/0912.4048

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