On The Stabilizing Index Of Hypergraphs

In 1960s Berge et al.introduced the notion of hypergraphs,which over-came the limitation of graphs on modeling binary relations.As hypergraphs can reflect the complicated multiple relationships in the real world,they play important roles in the fields of natural sciences and social sciences.Qi and Lin proposed the concept of eigenvalues of higher-order tensors in 2005.In 2012,Cooper and Dutle introduced the adjacency tensor of the uniform hypergraph,and extended some conclusions in the simple graph theory to the uniform hy-pergraph.Since then the spectral theory of hypergraphs has been an new area of algebraic graph theory.Tensor is an important tool for studying hypergraphs.From the gen-eralization of P-F theorem on nonnegative tensor,we know that()is an eigenvalue ofassociated with a nonnegative eigenvector.Ifis still weakly irreducible,the spectral radius()is associated with a unique pos-itive eigenvector up to a scalar,called the Perron vector.Differ from matri-ces,for the nonnegative weakly irreducible tensor,except for the Perron vector,there may be other eigenvectors corresponding to the spectral radius().Therefore,it naturally arises the problem of characterizing the number of eigenvectors associated with().Fan et al.defined the stabilizing index of the tensorand answered this question.In particular,ifis still symmetric,the calculation formula for the stabilizing index can be obtained from the Smith normal form of the corresponding incidence matrix over Z₈).Moreover,they applied it to the adjacency tensors of connected uniform hypergraphs,defined the stabilizing index of connected uniform hypergraphs,and gave some explicit formulas for the stabilizing indices of the hypergraphs with specified structures.The stabilizing index is an important parameter that reflects the struc-tures and properties of hypergraphs.In this thesis we mainly study the explicit formulas for the stabilizing index of uniform hypergraphs with some specified structures.The thesis is organized as follows.In the first chapter we mainly in-troduce the background,the main problems and our results.In the second chapter we introduce some basic knowledge,including some results which will be used in the following discussion.In the third chapter we mainly study the stabilizing indices of two classes of connected hypergraphs.The first class of hypergraphs is the coalescence₁()?₂()of two8)-uniform hypergraph-s₁,₂,whose stabilizing index(₁()?₂())is given in terms of the stabilizing indices(₁),(₂)of₁and₂respectively.By the result we can construct hypergraphs with sufficiently large stabilizing indices.The sec-ond class of hypergraphs is the Cartesian product₁2₂of two8)-uniform hypergraphs₁,₂.We give the formula of the incidence matrix₁₂₂of₁2₂,and obtain the formular of(₁2₂)by using the Smith normal form of₁₂₂over Z₈).In the fourth chapter we study the stabilizing index of the6)-folded coveringover a uniform hypergraph.We prove that in certain situation,the stabilizing index()ofdivides()of.