The minimum spectral radius of the r-uniform supertree having two vertices of maximum degree

A supertree is a connected and acyclic hypergraph. Let Tm△,△ be the set of the r-uniform supertrees with m edges and exactly two vertices of maximum degree ▵, where m≥ 5 and 3≤△≤(m+1)/2. We investigate the supertree with the minimum spectral radius among Tm△,△. Several useful grafting operations are introduced for comparing the spectral radii of supertrees. The supertree with the minimum spectral radius among Tm△,△ is characterized. In addition, a new simple method is demonstrated to obtain the supertree with the minimum spectral radius among the set of r-uniform supertrees with m edges.