Research On The Bounds Of Tensor Singular Vector Pairs And The Singular Values Radius Of Hypergraphs

Tensor theory is an important branch of mathematics and a powerful tool for solid mechanics and quantum physics.In recent years,the singular value problem of rectangular tensors has attracted much attention from scholars.It has important applications in quantum mechanics,and the research on the singular value of rectangular tensors has become a hot topic.Hypergraph is a general generalization of graph,the application of hypergraph to solve problems will be more specific and comprehensive.Hypergraphs are widely used in quantum chemistry,statistical mechanics,computer science,communication networks and information science.With the development of singular value theory of nonnegative rectangular tensors,many scholars try to study hypergraphs with nonnegative rectangular tensors.According to the Perron-Frobenius theorem of the nonnegative rectangular tensor given by K.C.Chang,L.Q.Qi et al in 2010,in this paper,we give some bounds on the principal ratios and the largest and smallest components of the pair of principal singular vectors of nonnegative irreducible rectangular tensors by using the ratios of arbitrary elements,the maximum row sum and the maximum column sum of tensors.Some bounds of the largest and smallest components of the principal singular vector pairs of the nonnegative irreducible rectangular tensors are given.As an application,some bounds of the singular value radius of uniformly connected hypergraphs and the largest and smallest components of the pair of principal singular vectors are given.