Study On The Spectrum Of Hypergraph And Its Extremal Structure

Since 2005,L.Qi and L.H.Lim defined the eigenvalue of the hypermatrix independently,and in 2012,J.Cooper and A.Dutle gave the definition of symmetric adjacency hypermatrix of n order k-uniform hypergraph,and the study of hypergraph problems based on hypermatrix has become a hot topic.The extreme value problem is a very challenging type of problem in graph theory.The extreme value problem of hypergraph based on hypermatrix has attracted the interest of many scholars.Related results have emerged continuously.At the same time,a large number of problems need to be further studied.Based on that,this paper mainly studies the spectrum and extremal structure of several types of k-uniform hypergraphs with given properties.In the first chapter,we first expound the main problems and background of the research,introduce the symbols,definitions and important lemmas which used in this article,then list the main research results of this article,and explain the framework structure of this article.In the second chapter,by constructing edge operations of the monotone property and using the technique of the weighted incidence matrix's ?-subnomal and power graph,the study obtains the maximum spectral radius of k-uniform unicyclic hypergraph with given diameter and its corresponding extremal structure.In the third chapter,by constructing edge operations of the monotone property and using the characteristic of the BFS-ordering,the study obtains the maximum spectral radius of k-uniform unicyclic hypergraph with given degree sequence and its corresponding extremal structure.In the fourth chapter,by constructing edge operations of the monotone property and using the technique of the weighted incidence matrix's ?-subnomal,power graph and symmetric hypermatrix computation,the study obtains the maximum spectral radius of k-uniform tricyclic hypergraph and its corresponding extremal structure.