Brauer-type Eigenvalues Inclusion Sets Of Tensors And Its Applications

Tensors were originally introduced by Gauss,Riemann and Christoffel et al.in the studies problems about geometrical space and multivariate differential theory of curved coordinate in the nineteenth century.In 2005,L.Qi and Lek-Heng Lim defined eigenvalues of tensors,respectively.L.Qi gave Ger?chgorin disk theorem of real symmetric tensors.Prof.K.C.Chang and Q.Yang et al.obtained Perron-Ferobenius theorem of nonnegative tensors,minimax theorem and some bounds of the Frobenius-type spectral radius of tensors.C.Li et al.gave some eigenvalue inclusion sets of tensors.Bu et al.obtained Brualdi-type eigenvalue inclusion sets of tensors and some bounds of spectral radius of weakly irreducible tensors via a digraph,respectively.Since then,eigenvalue problems of higher order tensors have become a hot and interesting topic in mathematics.This dissertation studies eigenvalue problems of tensors,the main results are as follows:(1)We give two Brauer-type eigenvalue inclusion sets theorems of tensors,some results on nonsingularity of tensors and spectral radius of tensors.Ger?chgorin disk domain in Qi's paper is proved bigger than the one in this dissertation.Results we obtained in this dissertation are more succinct and simple than existing results by comparing two Brauer-type eigenvalue inclusion sets theorems of tensors with Qi's via digital examples.(2)Brauer-type eigenvalue inclusion sets of matrix in matrix theory has been further generalized to tensors in this dissertation.The dissertation gives some Ky Fan-type theorems and some bounds of spectral radius of tensors.As application in hypergraph,there is a one to one correspondence between adjacent tensor of hypergraph and hypergraph.J.Cooper,X.Yuan,B.Zhou and H.Li et al.obtain some bounds of spectral radius of hypergraphs via parameters of graphs,such as degrees of vertices,diameters,the number of vertices and the number of edges.This dissertation gives some bounds of spectral radius of the adjacent tensor,Laplacian tensors and signless Laplacian tensors of hypergraphs by using skills of Brauer-type eigenvalue inclusion sets of tensors.The dissertation compares obtained results in the dissertation with conclusions in the references on theory and digital examples,and obtain some bounds on the spectral radius of tensors and hypergraphs in the dissertation,which arebetter than the existing conclusions.