Sufficient Conditions for Spanning k-Hypertrees via Distance Spectral Radius of Hypergraphs

For an integer k≥2, a spanning k-hypertree T is defined as a spanning hypertree (β-acyclic) such that the maximum degree dTv of every vertex v∈VT 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 λDH, is defined as the largest eigenvalue of its distance matrix DH. A lower bound for λDH 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.