On The Spectral Symmetry Of Uniform Hypergraphs

The simple graphs mainly deal with the binary relations of discrete ob-jects,or equivalently the system of the subsets of two elements of a finite set.As a generalization of simple graphs,the hypergraphs mainly deal with the system of the subsets of multiple elements of a finite set.The matrix is a main tool to study the spectra of simple graphs.As a generalization of matrix,the tensor is a powerful tool to study the spectra of hypergraphs,which also has widely applications in information theory,computer science and operation research etc.If the spectrum Spec(A)of a tensor A satisfies that Spec(A)= ei2π/lSpec(A),then A is called spectral l-symmetric.If the adjacency tensor A(G)of a unifor-m hypergraph G is spectral l-symmetric,then G is called spectral l-symmetric.In 2011 Cooper and Dutle give an expression of the characteristic polynomial of the adjacency tensor by the generalized traces,and pose a problem of char-actrizing the spectral k-symmetry of k-uniform hypergraphs.In 2013 Shao et.al give a graph interpretation and explicit formula of the generalized traces,and characterize the spectral k-symmetry of k-uniform hypergraphs.In 2014 Zhou et.al discuss the spectral 2-symmetry of k-uniform hypergraphs.In 2017 Nikiforov characterize the the spectral 2-symmetry of k-uniform hypergraphs by the odd coloring of hypergraphs.In this thesis we mainly discuss the problem of how to characterize the general spectral symmetry of k-uniform hypergraphs,i.e.spectral l-symmetry.We give a characterization of the spectral l-symmetry of a general tensors of order k,and get some results on the spectral symmetry of hypergraphs by applying the result.We also discuss four classes of uniform hypergraph:p-hm bipartite hypergrpahs,odd-bipartite generalized power hypergraphs,non-odd-bipartite generalized power hypergraphs,and a class of 3-uniform hypergraphs on torus,and obtain some results on spectral symmetry.The organization of the thesis is as follows.In Chapter one,we briefly in-troduce the study on hypergraphs and tensors,some basic formulas,concepts and notions,and the development of the spectral symmetry on hypergrpah-s and the main results in this thesis.In Chapter two,we introduce some preliminaries including the expression of the characteristic polynomial of the adjacency tensor by generalized traces,and the Perron-Frobenius theorem of nonnegative tensors.In Chapter three,we give the main results of the thesis,including the characterization of spectral l-symmetry of a general tensor of order k,the characterization of spectral symmetry of p-hm bipartite hypger-graphs,the cyclic index of odd-bipartite or non-odd-bipartite generalized pow-er hypergraphs,and the characterization of a class of 3-uniform hypergraphs on torus.