On The Spectral Radii Of Uniform Hypergraphs

The graph is an very important model for studying the structure of the system consisting of discrete objects with binary relation. As a generalization, the hyper-graphs are the system of the subsets of a finite set, which are the better model for characterizing the complex multi-array relation for the real world.Fan Chung, Feng, Li, Friedman, Wigderson, Rodriguez, Lu and Peng et. al introduced the adjacency or Laplacian matrix of a hypergraph to investigate its property. However, as the edges of a hypergraph are determined by more than two vertices, the above definition can not reflect the structural property of the hyper-graph directly. Lek-Heng Lim, Liqun Qi and Gongqing Zhang all introduced the eigenvalues and eigenvectors of a hypergraph and the relation for their. Since then the eigenvalues and eigenvectors of the tensors (such as adjacency tensor, Laplacian tensor and signless Laplacian tensor) associated with a uniform hypergraph have been studied extensively.In 2014, Honghai Li, Liqun Qi and Jiayu Shao characterized the maximum or minimum spectral radius of the power of trees. In this paper we discuss a general problem, that is, to characterize the maximum spectral radius among all hypergraphs with given number of edges. We proved that among all κ-uniform hypergraphs with given number of vertices and edges, the hypergraph with maximum spectral radius contains a vertex adjacent to all other vertices. We characterized the maximizing unicyclic hypergraph, linear or power unicyclic/bicylic hypergraphs, respectively.The organization of the thesis is as follows. In Chapter 1 we briefly introduce the development of spectral theory of graphs and hypergraphs, some concepts and notations, and the problem and main results in this thesis. In Chapter 2 we first introduce the Perron-Erobenius theorem of nonnegative tensors, and then charac-terize the change of the spectral radius of a hypergraph after one of its branches is relocated. In Chapter 3 we characterized the maximizing unicyclic hypergraph, linear or power unicyclic/bicylic hypergraphs, respectively.