2009/12/20 by Sylvain E. Cappell, Cappell, Sylvain E., Edward Y. Miller +1
Computer Science · Mathematics · Physics and Astronomy · #05C85 #15A42 #35P15 #46N50 #52B60 #57M15 #68R10 #94C15 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Quantum chaos and dynamical systems #Quantum optics and atomic interactions #Spectral Theory in Mathematical Physics #cs.DM #math.CO #msc:05C85 #msc:15A42 #msc:35P15 #msc:46N50 #msc:52B60 #msc:57M15 #msc:68R10 #msc:94C15
paper · pdf · doi:10.48550/arxiv.0912.4048
arxiv created 2009/12/20 · openalex publication_date 2009/12/20 · arxiv updated 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.