On The Least H-eigenvalue Of Hypergraphs

Simple graphs are the systems of 2-subsets of finite sets,which describe the binary relations of finite objects.Matrices play an important role on char-acterizing the structural properties of simple graphs,which yields the spectral graph theory.In order to describe the multi-relations of finite objects,the no-tion of hypergraphs are introduced,which are the systems of subsets of finite sets.Since Liqun Qi and Lek-Heng Lim introduced the eigenvalues of tensors independently in 2005,researchers apply the adjacency tensors to represen-t the hypergraphs,and use their eigenvalues to characterize the structure of hypergraphs.The spectra of hypergraphs have been a hot topic in algebraic graph theory and spectral graph theoryChang,Yang and Friedland et al.generalized the Perron-Frobenius from nonnegative matrices to nonnegative tensors.By the theorem,for a connect-ed k-uniform hypergraph G,the spectral radius ?(G)of the adjacency tensor A(G)is an eigenvalue of A(G)corresponding to a unique positive eigenvector up to a positive scalar.As the eigenvalues of tensors may be complex num-bers or only associated with complex eigenvectors,people pay attention to H-eigenvalues,i.e.the eigenvalues associated with a real eigenvector.Surely,the spectral radius of hypergraph is the largest H-eigenvalue,and hence is an important content in spectral hypergraph theoryComparing with the spectral radius,the least H-eigenvalue of A(G)re-ceives little attention.Denote by Amin(G)the least H-eigenvalue of A(G)for a connected k-uniform hypergraph G,where k is even.In 2014,Nikiforov gave some lower bounds of ?min(G)in terms of the order and size of G.Shao et al.proved that-?(G)is the(least)H-eigenvalue of G if and only if k is even and G is odd-bipartite.So,for odd-bipartite hypergraphs,the problem on the least H-eigenvalues is equivalent to that on the spectral radius.In 2016 Fan et al.constructed a series of non-odd-bipartite hypergraphs whose least H-eigenvalues convergent to(?).For non-odd-bipartite hypergraphs G,-?(G)is not the H-eigenvalue of G but may be the eigenvalue of G(the N-eigenvalue).In 2017 Nikiforov proved that-?(G)is an eigenvalue of G if and only if G is odd-colorable.Nikiforov and Fan et al.gave some examples of odd-colorable but non-odd-bipartite hypergraphs.In 2019,Fan et al.presented a systematic characterization of odd-bipartiteness and odd-colorability.In this thesis we mainly study the least H-eigenvalues of two classes of hypergraphs,i.e.the hypergraphs with cut vertices and the products of hy-pergraphs.In Chapter two we firstly give an upper bound of the least H-eigenvalue,and then characterize the property of the first eigenvectors(the real eigenvectors corresponding to the least H-eigenvalue).We obtain the per-turbation result of the least H-eigenvalue of a hypergraph with cut vertices under one branch relocating from one vertex to another vertex.By the per-turbation result,we characterize the unique hypergraph which minimizes the least H-eigenvalue among all hypergraphs in a certain class.In Chapter three,we consider the least H-eigenvalues of Cartesian product G?H and direct product G x H.By the incidence matrix equation over Z2,we prove that G?H is odd-bipartite if and only if G and H are both odd-bipartite.We show that ?min(G?H)-?Amin(G)+?min(H)For the direct product of hypergraphs,we prove that G ×H is odd-bipartite if one of G and H is odd-bipartite;and in this case?min(G × H)=-(k-1)!p(G)p(H).In a general case,?min(G x H)<min{(k-1)!?min(G)?(H),(k-1)!?(G)?min(H)}.