Geometry and spectrum of rapidly branching graphs |
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| Authors: | Matthias Keller Florentin Münch Felix Pogorzelski |
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| Affiliation: | 1. Institut für Mathematik der Universit?t Potsdam, Potsdam OT Golm, Germany;2. Mathematisches Institut, Friedrich Schiller Universit?t Jena, Jena, Germany;3. Department of Mathematics, Technion ‐ Israel Institute of Technology, Haifa, Israel |
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| Abstract: | We study graphs whose vertex degree tends to infinity and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness. |
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| Keywords: | Graph Laplacians discrete spectrum eigenvalue asymptotics isoperimetric estimates stochastic completeness 05C63 39A70 58J50 |
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