Cayley树上Laplacian特征向量和特征值的显式构造
Explicit construction of the eigenvectors and eigenvalues of the graph Laplacian on the Cayley tree
摘要：
对任意图的广义傅里叶分析要求对图的拉普拉斯算子的特征向量有详细的了解。利用Cayley树的对称性，我们递归地构造具有指数增长特征空间的特征向量族，与谱的下部的特征值相关联。对于大树，谱隙随树的大小呈指数衰减。特征值和特征向量服从由树的嵌套几何形状产生的递归关系。非周期网络本征向量的这种解析解是为我们早先提出的谱重整化群提供坚实的基础所必需的。Tuncer和A. Erzan，Phys. Rev. E { \bf 92}，022106（2015）]。PACS编号02.10.Ox组合学;图论，02.10.Ud线性代数，02.30 Nw傅立叶分析
英文摘要：
A generalized Fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph Laplacian. Using the symmetries of the Cayley tree, we recursively construct the family of eigenvectors with exponentially growing eigenspaces, associated with eigenvalues in the lower part of the spectrum. The spectral gap decays exponentially with the tree size, for large trees. The eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree. Such analytical solutions for the eigenvectors of non-periodic networks are needed to provide a firm basis for the spectral renormalization group which we have proposed earlier [A. Tuncer and A. Erzan, Phys. Rev. E {\bf 92}, 022106 (2015)]. PACS Nos. 02.10.Ox Combinatorics; graph theory, 02.10.Ud Linear algebra, 02.30 Nw Fourier analysis
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