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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2406.17965 |
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| _version_ | 1866908284621946880 |
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| author | Harrison, Jonathan Weyand, Tracy |
| author_facet | Harrison, Jonathan Weyand, Tracy |
| contents | Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. 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 [20]. 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 [20]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Determinants of Almost Equilateral Quantum Graphs Harrison, Jonathan Weyand, Tracy Mathematical Physics Spectral Theory 35A01, 65L12, 65L20, 65L70 Kirchoff's matrix tree theorem of 1847 connects the number of spanning trees of a graph to the spectral determinant of the discrete Laplacian [22]. 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 [20]. 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 [20]. |
| title | Spectral Determinants of Almost Equilateral Quantum Graphs |
| topic | Mathematical Physics Spectral Theory 35A01, 65L12, 65L20, 65L70 |
| url | https://arxiv.org/abs/2406.17965 |