The Spectral Properties On The Tensor Associated With An Uniform Hypergraph

The spectral theory of graphs is an important branch of the graph theory and its applications, in which it is well known that the matrix services as an important tool. Tensors as the general form of matrices have been found extensive applications in mathematics, physics, mechanics, etc. Moreover, tensor is also an important tool for the research on tensor spectral theory of uniform hypergraphs.In this thesis, we further investigated the properties of the largest H-eigenvalue of the adjacency tensors, Laplacian and signless Laplacian tensors of uniform hy-pergraphs and get many meaningful results. Besides, we generalize the spectral bounds on the clique number of graphs to that of uniform hypergraphs.This thesis consists of three parts. Firstly, we study the properties of H-eigenvalues of the adjacency tensor related to uniform hypergraphs, including hyperstar and odd-bipartite hypergraph, and get the new upper and lower bounds on the largest H-eigenvalue of the adjacency tensor.In the second part, firstly, we study the properties of the largest H-eigenvalue of Laplacian tensor. We proved that the number of linearly independent H+-eigenvector corresponding to 0 equal to the number of components of uniform hypergraphs, and give a new upper bound on the largest H-eigenvalue of Lapla-cian tensor. Secondly, we establish the relationship of the largest H-eigenvalue between the adjacency tensor and the signless Laplacian tensor of uniform hyper-graphs. Moreover, we also give the new upper and lower bounds on the largest H-eigenvalue of the signless Laplacian tensor of uniform hypergraphs.Thirdly, we study the spectral bounds on the clique number of uniform hy-pergraphs. Given a uniform hypergraph, we use the largest H-eigenvalue of the adjacency tensor to express the bounds on the clique number of the uniform hypergraph, and get the upper and lower bounds on the clique number.