The maximum α-spectral radius and the majorization theorem of k-uniform supertrees

Let α be a real number with 0 ≤ α 1 , G be a uniform hypergraph, and A α ( G ) = α D ( 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 A α ( 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.