The Spectrum Of Some Hypergraphs

In a simple graph G = (V ,E), let the matrix A( G) be the adjacency matrix of G , so the eigenvalues of A( G) are named the eigenvalues of G . The sequence made up of all the eigenvalues is called the spectrum of G . All the time, the estimate of the eigenvalues of graphs is a quite active question for discussion. Over the past 30 years, the articles on the graph problem have been plentiful. The study of the coincidence relation between the spectral distribution of graph and the structure of a graph can not only hasten the depict of the interrelationship of discrete structure in theory, but also have profound practical background in application, such as the network's optimizes and designs, the design of integrated circuit, and operations research.Hypergraph are set system of finite sets, they play an important role in the discrete mathematics. In the early times, the research of hypergraph has been spreaded the classic conclusion of graph. Owing to more and more presentation of results, scholars have paid more attention to the hypergraph. Because the adjacency matrix of hypergraph and graph are different things, the research of hypergraph's spectrum can't continue to use the method of graph. In the paper[4], the author mentioned that getting point-edge bipartite graph first, the adjacency matrix's entry of hypergraph depend on the length of point-edge bipartite graph's two points, which come from the point set of hypergrph. In fact, the definition of the spectrum of hypergraph could not reflect reasonly the relationship among the points of hypergraph.In this paper, we use polynomial to present hypergraph, the spectrum of hypergraph can get from the extreme value of polynomial under restrict condition.In the first part, introduce relevant concept and term on the spectrum of a graph and the spectrum of hypergraph.In the second part, We have obtained some more important conclusions. For example, we can describe the spectrum of 3-uniform star hypergraph by the spectrum of graphs from theorem 2.13. Namely, we can obtain the spectrum of graphs by adjacency matrix, then we obtain the relation between the the spectrum of 3-uniform star hypergraph and the spectrum of graphs in order to compute the the spectrum of 3-uniform star hypergraph,further more ,we research the spectrum of 3-uniform star hypergraph.