Characterizations Of K-cyclic Linearly Uniform Hypergraphs With Maximal Spectral Radius

Graph is an effective mathematical model for studying the binary relation of discrete objects.A hypergraph is a general generalization of an ordinary graph.An edge of a graph contains two vertices,while an edge of a hypergraph contains multiple vertices.Hypergraphs can better reflect the complex and multiple relationships among real objects.It is an important topic in the theory of spectral extremum of hypergraphs to characterize the structure of hypergraphs with extremal spectral radius by analyzing the influence of moving edges and other operations on the spectral radius of hypergraphs.This paper mainly study the3-cyclic and k(k(29)3)-cyclic linear hypergraphs with the maximum spectral radius by using the relocating edges theorem and the properties of weighted incidence matrix.First,the properties of connected graphs with the largest spectral radius in the graph class with m edges are given,and the structures of the 3-cyclic and k-cyclic power hypergraph with the maximum spectral radius are described by using the relation between the spectral radius of graph and the spectral radius of the power hypergraph.Secondly,the 3-cyclic and k-cyclic linear non-power hypergraphs with maximum spectral radius are described by using the relocating edges theorem and the related properties of weighted incidence matrix.Then,the obtained hypergraphs are compared according to the definition of tensor eigenvalues and the 3-cyclic and k-cyclic linear hypergraphs with the maximum spectral radius are described.Finally,some conjectures about the characterization of k-cyclic linear uniform hypergraphs are given.