The maximum [formula omitted]-spectral radius and the majorization theorem of [formula omitted]-uniform supertrees

Abstract

Let α be a real number with 0≤α<1, G be a uniform hypergraph, and Aα(G)=αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal degree tensor and the adjacency tensor of G, respectively. The spectral radius of Aα(G) is called the α-spectral radius of G. In this paper, some properties for the α-spectral radius of connected k-uniform hypergraphs are investigated, the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with a given degree sequence is characterized, and the majorization theorem for k-uniform supertrees is obtained. By applying the majorization theorem, we determine the k-uniform supertrees obtaining the three largest α-spectral radius among all k-uniform supertrees with n vertices, give a new proof ordering the eight largest spectral radius among all k-uniform supertrees with n vertices, propose an open problem for further research, and characterize the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with given different parameters, e.g., the maximum degree △, or the number of pendent vertices.