一类球齐次图的谱分析
Spectral Analysis of Certain Spherically Homogeneous Graphs
摘要：
研究了具有某种球面齐性的有根图上的算子。这些图被称为路径交换，并且允许将邻接矩阵和拉普拉斯算子分解为反映图的结构的雅可比矩阵的直和。因此，邻接矩阵和拉普拉斯算子的谱性质可以通过Jacobi矩阵的详细理论来分析。对于一些例子，其中包括反树，我们推导出明确的分解，并提出了一个动物园的光谱行为引起的几何图形。特别是，这些例子表明，谱型在粗糙等距下根本不稳定。
英文摘要：
We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the structure of the graph. Thus, the spectral properties of the adjacency matrix and the Laplacian can be analyzed by means of the elaborated theory of Jacobi matrices. For some examples which include antitrees, we derive the decomposition explicitly and present a zoo of spectral behavior induced by the geometry of the graph. In particular, these examples show that spectral types are not at all stable under rough isometries.
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