论文标题
可以听到量子图的跨越树吗？
Can One Hear the Spanning Trees of a Quantum Graph?
论文作者
论文摘要
基尔乔夫（Kirchhoff）表明，图的跨树木数量是组合拉普拉斯（Laplacian）除以顶点的数量的频谱决定因素。我们在量子图设置中重新构架此结果。我们证明，Laplace操作员在有限连接的度量图上具有标准（Neummann-Kirchhoff）顶点条件的光谱决定因素确定当度量图的边缘长度足够接近时，确定了跨越树的数量。为了获得此结果，我们分析了一个等边量子图，该量子图与离散图运算符的光谱密切相关，然后在边缘长度的扰动下使用光谱决定因素的连续性。
Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral determinant of the Laplace operator on a finite connected metric graph with standard (Neummann-Kirchhoff) vertex conditions determines the number of spanning trees when the lengths of the edges of the metric graph are sufficiently close together. To obtain this result, we analyze an equilateral quantum graph whose spectrum is closely related to spectra of discrete graph operators and then use the continuity of the spectral determinant under perturbations of the edge lengths.
