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Abstract
For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on onesided infinite nonbacktracking paths. Interpreting the transfer operator as a classical dynamical system and the Laplace operator as its quantization, this result can be viewed as a quantumclassical correspondence. In contrast to previously established quantumclassical correspondences for the vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence is valid for all parameters. This allows us to relate certain spectral quantities to topological properties of the graph such as the cyclomatic number and the 2colorability. The quantumclassical correspondence for the edge Laplacian is induced by an edge Poisson transform on the universal covering of the graph which is a tree of bounded degree. In the special case of regular trees, we relate both the vertex and the edge Poisson transform to the representation theory of the automorphism group of the tree and study associated operator valued Hecke algebras.
Originalsprog  Engelsk 

Antal sider  46 
DOI  
Status  Afsendt  15 dec. 2023 
Fingeraftryk
Dyk ned i forskningsemnerne om 'Edge Laplacians and Edge Poisson Transforms for Graphs'. Sammen danner de et unikt fingeraftryk.Projekter
 1 Igangværende

Spectral Correspondences for Trees and Buildings
Frahm, J. (PI) & Arends, C. (Deltager)
01/04/2023 → 31/03/2026
Projekter: Projekt › Forskning