Some Bounds On The Spectral Radii Of Nonnegative Tensors

The concept of tensor is originated in the 19 th century.Recently,tensor has many important applications in the areas of theoretical physics,magnetic resonance imaging,quantum mechanics,higher order Markov chains.In 2005,Qi and Lim gave the concept of eigenvalue on tensors,independently.In 2008,Chang et al.proved the Perron-Frobenius theorem for nonnegative tensors.In 2010,Yang et al.researched some properties of the spectral radius for nonnegative tensors.In 2013,Friedland et al.introduced the notion of weakly irreducible tensors,and showed the Perron-Frobenius theorem for nonnegative weakly irreducible tensors.Recently,due to the development of spectral theory of tensors,spectral hypergraph theory has develop rapidly.In 2012,Cooper et al.gave the concept of the adjacency tensor of a uniform hypergraph.In 2014,Qi defined the Laplacian tensor and the signless Laplacian tensor of a uniform hypergraph.The problem on eigenvalue of tensor has got much attention.This paper is organized as follows.When A is nonnegative weakly irreducible tensor,the spectral radius of A is an eigenvalue of A.we obtain the generalized Frobenius-type bounds on the spectral radius of nonnegative tensor A by the product of diagonal matrix and A,and we gave the bounds of the spectral radius for nonnegative tensor in terms of slice sum and parameter ?,and prove the result of this paper is better.For a nonnegative weakly irreducible tensor A,we present the generalized Brauer-type inequalities on the spectral radius of A via subtensor of A.The relation between eigenvalues and structure of graph are the core problem of spectral graph theory.It is natural to generalize spectral theory to hypergraph.Recently,the spectral hypergraph theory has received much attention.As applications,some bounds on the spectral radius of hypergraphs are presented,for some examples we prove the results of this paper are better than the known ones.