The Minimal 3-uniform Hypergraphs With Stabilizing Index 1

As a natural generalization of simple graphs,the hypergraphs have deeper theoretic value and wider application.The spectrum of the adjacency matrix of a simple graph plays an important role in describing its structural prop-erty.When investigating the structure of a hypergraph,the spectrum of the adjacency tensor will play an important role.Let A be a nonnegative tensor of order m and dimension n.By the Perron-Frobenius theorem of nonnegative tensor,the spectral radius p(A)of A is an eigenvalue of A,corresponding to a nonnegative eigenvector.If A is nonnegative weakly irreducible,then p(A)corresponds a unique positive eigenvector(called Perron vector),and any positive eigenvector corresponds to p(A).When m = 2,A is nonnegative irreducible matrix,and there is a unique eigenvector corresponds to ?(A)up to a scalar,i.e.the Perron vector However,when m ? 3,i.e.A is a nonnegative weakly irreducible tensor of high order,there are more than one eigenvector corresponding to p(A).Fan et.al introduced the stabilizing index of tensors,and proved that the stabilizing index of a nonnegative weakly irreducible tensor is exactly the number of eigenvectors corresponding to the spectrum radius.For a symmet-ric nonnegative weakly irreducible tensor of order m,they applied the Smith norm form of the incidence matrix over Zm to give an explicit formula of the stabilizing index.In addition,they characterized the 3-uniform hypergraph with stabilizing index greater than 1.This thesis mainly investigates the 3-uniform hypergraphs with stabiliz-ing index 1.By using the stabilizing labeling of the hypergraphs,we give an equivalent characterization of the hypergraphs with stabilizing index 1,togeth-er with some structural properties.Based on the those structural properties,we introduce minimal uniform hypergraphs,that is,the hypergraphs with sta-bilizing index 1 but any proper subhypergraphs with stabilizing index greater than 1.We characterize the minimal 3-uniform hypergraph,and describe the structural property of a minimal 3-uniform hypergraph when its maximum degree is n-1,n-2,or 3.The thesis is organized as follows.In Chapter one,we briefly introduce the the spectral study of hypergraphs and tensors,some basic formulas,con-cepts and notations,the problems and the main results we obtained in this thesis.In Chapter two,we introduce some preliminary knowledge,including the Perron-Frobenius theorem of nonnegative tensors,the stabilizing index and the stabilizing labelling of hypergraphs.In Chapter three,we first char-acterize the structure of hypergraphs with stabilizing index 1,then introduce the minimal 3-uniform hypergraphs and give an equivalent characterization for the minimal 3-uniform hypergraphs,finally we characterize the structural property of the minimal 3-uniform hypergraphs when the maximum degrees are n-1,n-2,3 respectively.