تقرير
Spectral Determinants of Almost Equilateral Quantum Graphs
| العنوان: | Spectral Determinants of Almost Equilateral Quantum Graphs |
|---|---|
| المؤلفون: | Harrison, Jonathan, Weyand, Tracy |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematical Physics, Mathematics - Spectral Theory, 35A01, 65L12, 65L20, 65L70 |
| الوصف: | Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [6]. Recently an analogue was obtained for quantum graphs relating the number of spanning trees to the spectral determinant of a Laplacian acting on functions on a metric graph with standard (Neumann-like) vertex conditions [11]. This result holds for quantum graphs where the edge lengths are close together. A quantum graph where the edge lengths are all equal is called equilateral. Here we consider equilateral graphs where we perturb the length of a single edge (almost equilateral graphs). We analyze the spectral determinant of almost equilateral complete graphs, complete bipartite graphs, and circulant graphs. This provides a measure of how fast the spectral determinant changes with respect to changes in an edge length. We apply these results to estimate the width of a window of edge lengths where the connection between the number of spanning trees and the spectral determinant can be observed. The results suggest the connection holds for a much wider window of edge lengths than is required in [11]. Comment: 17 pages, 2 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.17965 |
| رقم الأكسشن: | edsarx.2406.17965 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |