公共交通网络上的超巨星光谱
Spectra of hyperstars on public transportation networks
摘要：
本文的目的是引入一个模型来研究在公共交通网络中广泛存在的结构。我们表明，通过超图，可以描述这些结构和调查它们的光谱之间的关系。为此，我们将图上的$(m,k)$ -星的结构推广到超图上：超图上的$(m,k)$ -超星。通过给出适当的超边权条件，证明了可计算值和重数的矩阵特征值的存在性，其中所考虑的矩阵是Laplacian矩阵、邻接矩阵和转移阵.分别考虑一般超图和一致超图的情况，证明了具有$(m,k)$ -超星的超图的两种顶点集约简是可行的，且保持相同的特征值和约简的重数.最后，一些有用的特征向量的性质，推导出一个产品与一个合适的矩阵，我们将这些结果与超图上的Fiedler谱划分。
英文摘要：
The purpose of this paper is to introduce a model to study structures which are widely present in public transportation networks. We show that, through hypergraphs, one can describe these structures and investigate the relation between their spectra. To this aim, we extend the structure of $(m,k)$-stars on graphs to hypergraphs: the $(m,k)$-hyperstars on hypergraphs. Also, by giving suitable conditions on the hyperedge weights we prove the existence of matrix eigenvalues of computable values and multiplicities, where the matrices considered are Laplacian, adjacency and transition matrices. By considering separately the case of generic hypergraphs and uniform hypergraphs, we prove that two kinds of vertex set reductions on hypergraphs with $(m,k)$-hyperstar are feasible, keeping the same eigenvalues with reduced multiplicity. Finally, some useful eigenvectors properties are derived up to a product with a suitable matrix, and we relate these results to Fiedler spectral partitioning on the hypergraph.
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