A supertree is a connected and acyclic hypergraph. Denote by T_m,n,α the set of m‐uniform supertrees of order n with independent number α. Focusing on the spectral radius in T_m,n,α, this present completely determines the hypergraphs with maximum spectral radius among all the supertrees with n vertices and independence number α for [m − 1/mn] ≤ α ≤ n − 1, which extend the results of Lu et al. from tree to uniform supertree. Our techniques are based on the structure properties of supertrees with given independence number and general edge‐moving operation. As a byproduct, we also determine the hypergraphs with minimum signless Laplacian spectral radius in T_m,n,α.

In this paper, we defined two classes of hypergraphs, hyperbugs and kite hypergraphs. We show that balanced hyperbugs maximize the spectral radii of hypergraphs with fixed number of vertices and diameter and kite hypergraphs minimize the spectral radii of hypergraphs with fixed number of vertices and clique number.

The relationship between the tensor spectrum and structural parameters of hypergraphs is a hot topic in the field of hypergraph theory in recent years. This work first establishes the relation between the $ \alpha $-spectral radius and diameter for $ s $-uniform hypergraphs of a given order. Second, we obtain the relation between $ \alpha $-spectral radius, connectivity and independence number of $ 3 $-uniform hypergraphs with given order. Additionally, the extremal structure achieving the maximum $ \alpha $-spectral radius is characterized for supertrees, specifically through the difference between the maximum degree and its $ \alpha $-spectral radius.

For an integer k ≥ 2, a spanning k ‐hypertree T is defined as a spanning hypertree ( β ‐acyclic) such that the maximum degree d _T ( v ) of every vertex v ∈ V ( T ) is at most k . A sufficient condition for the existence of a spanning k ‐hypertree in connected hypergraphs is established. The algorithm for checking the existence of a spanning k ‐hypertree and its complexity analysis are also presented. The distance spectral radius of a hypergraph H , denoted as λ ( D ( H )), is defined as the largest eigenvalue of its distance matrix D ( H ). A lower bound for λ ( D ( H )) is established for a connected hypergraph H . Combined with typical distance spectral techniques and structural analysis of hypergraphs, this bound yields sufficient conditions for the existence of spanning k ‐hypertrees in terms of the distance spectral radius.