Spectral determinants of almost equilateral quantum graphs

IF 1.4 3区 数学 Q1 MATHEMATICS
Jonathan Harrison, Tracy Weyand
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引用次数: 0

Abstract

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].

几乎等边量子图的谱行列式
1847年Kirchoff的矩阵树定理将一个图的生成树的数目与离散拉普拉斯[22]的谱行列式联系起来。最近得到了一个量子图的类比,将生成树的数目与作用于具有标准(类诺伊曼)顶点条件的度量图上的函数的拉普拉斯算子的谱行列式联系起来。这个结果适用于边长接近的量子图。边长相等的量子图称为等边图。这里我们考虑等边图,其中我们扰动了一条边的长度(几乎是等边图)。我们分析了几乎等边完全图、完全二部图和循环图的谱行列式。这提供了光谱行列式随边缘长度变化的速度的度量。我们应用这些结果来估计边缘长度窗口的宽度,其中可以观察到生成树数量和谱行列式之间的联系。结果表明,该连接保持了比[20]所需的更宽的边缘长度窗口。
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来源期刊
Analysis and Mathematical Physics
Analysis and Mathematical Physics MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.70
自引率
0.00%
发文量
122
期刊介绍: Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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