Research On The Spectral Radii And Principal Eigenvectors Of Nonnegative Tensors

Tensor theory is an important branch of mathematics and has important applications in mechanics and physics.In recent years,following the rise of quantum computing,machine learn and artificial intelligence,some new problems in tensor theory have attracted the attention of scholars,such as eigenvalues of tensors,spectral theory of hypergraph via tensors and so on.In 2005,Professor Liqun Qi from Hong Kong Polytechnic University and Professor Lek-Heng Lim from University of Chicago gave the definition of eigenvalue on tensor,respectively.In 2008,Professor Kungching Chang from Peking University systematically extended the nonnegative matrix Perron-Frobenius theory to nonnegative tensor.Recently,due to the development of spectral theory of tensor,many scholars have begun to use tensor to study spectral hypergraph theory.In 2012,J.Cooper and A.Dutle gave the concept of the adjacency tensor of uniform hypergraph.Liqun Qi gave the definitions of Laplacian tensor and signless Laplacian tensor of uniform hypergraph in 2014.After that,the problem on eigenvalues of tensor has got much attention.In 2012,a member of the Russian Academy of Science Kolotilina described the bounds of spectral radius for nonnegative matrix using the associated digraph on it.In this paper,we generalize this result to nonnegative tensor and study some results of the principal eigenvector of nonnegative weakly irreducible tensor,this paper is organized as follows.We obtain some bounds for the spectral radius of nonnegative weakly irreducible tensor ? by using the associated digraph on it.For nonnegative weakly irreducible tensor,the spectral radius ?(?)is an eigenvalue of ? and there exists unique a positive eigenvector corresponding to it.The ratio of the maximum and minimum entries of this positive eigenvector is called the principal ratio of ?.We give a lower bound of the principal ratio and some bounds on the maximum and minimum entries of the principal eigenvector for nonnegative weakly irreducible tensor.As applications,we obtain some bounds for the principal ratio and the entries on the principal eigenvector of the signless Laplacian tensor for connected uniform hypergraph.The relation between eigenvalues on tensor and structure of hypergraph are the core problem of spectral hypergraph theory.Recently,the spectral hypergraph theory has received much attention.We also obtain some bounds for the spectral radius of the signless Laplacian tensor via some graph parameters,such as degrees of vertices,sizes of edges and diameter for connected uniform hypergraph.