On The Cyclic Index Of Hypergraphs

As a generalization of simple graphs,hypergraphs can describe the multi-relations in the real world,and play an important role in many fields such as complex network,biology network and data structure.In 2005 Liqun Qi and Lek-Heng Lim introduced the eigenvalues of tensors or hypermatrices indepen-dently.In 2012 Cooper and Dutle gave the tensor representation of uniform hypergraphs by the adjacency tensors.Since then the spectra of hypergraphs have been an hot topic in algebraic graph theory and spectral graph theory.In 2019 Fan et al.studied the spectral symmetry of tensors in a general viewpoint,that is,the invariant property the spectrum of a tensor under a certain rotation of complex plane.Formally,if the spectral Spec(A)of a tensorA satisfies that Spec(A)=(?)Spec(A),then A is called spectral l-symmetric,and the maximum integer l satisfying the above equality is called the cyclic index of A.A uniform hypergraph is spectral l-symmetric(or has cyclic index c(G)),if its adjacency tensor A(G)is spectral l-symmetric(or has cyclic index c(G)).For some special situations,Cooper and Dutle proved that an m-partite hypergraph is spectral m-symmetric,and posed the problem of characterizing the spectral m-symmetry of m-uniform hypergraphs in 2012.In 2015,Shao et al.characterized the spectral m-symmetry by using the generalized traces of tensors.In 2014 Pearson and Zhang posed the problem of characterizing the symmetric spectrum of m-uniform hypergraphs,i.e.the spectral 2-symmetry.In 2015 Shao et al.characterized the symmetric H-spectra of hypergraphs,and asked whether the symmetric H-spectrum is equivalent to symmetric spectrum.Fan et al.gave a negative answer to the above problem by constructing a class of non-odd-bipartite generalized power hypergraphs in 2015.Nikiforov completely characterized the symmetric spectra of hypergraphs by introducing the odd-coloring of hypergraphs in 2017In this thesis we mainly study the cyclic indices of two classes of hyper-graphs,i.e.the generalized power hypergraphs and the products of hyper-graphs.Fan et al.posed the following conjectureConjecture:Let G be a t-uniform hypergraph,and let Gm,s be a gener-alized power of G with m=st.Then c(Gm,s)=s·c(G)In Chapter two of the thesis,we gave a negative answer to the Conjecture by using Nikiforov’s hypergraph as a counterexample.We show that Gm,s isspectral s·c(G)/(s,(c(G))-symmetric.which implies that the conjecture holds in some cases such as(s,(c(G))=1 or(s,t)=1.We give an equivalent condition for the equality in the Conjecture by the incidence matrix equation over Zm.In Chapter three of the thesis,we consider two types of products of hy-pergraphs,i.e.the Cartesian product G□H and the direct product G x H,and establish the relationship between the cyclic index of the product and those of its factors.We give an explicit formula of the incidence matrix of G□H.By using the incidence matrix equation over Zm,we get the cyclic index of G□H,that is,c(G□H)=(c(G),c(H)).We prove that is spectral[(c(G),c(H)]-symmetric,i.e.[c(G),c(H)]| c(G × H),where(a,b),[a,b]denote the greatest common divisor and the least common multiple of a,b respectively.