The spanning $k$-hypertree and adjacency spectral radius in hypergraphs

Abstract

The spanning $k$-tree of a connected graph is a spanning tree with a maximum degree not exceeding $k$. Hypergraphs provide a natural framework for modeling higher-order interactions. It is natural to extend the concept of spanning $k$-trees from graphs to hypergraphs. Tensor-based spectral models for hypergraphs are inherently higher-order and nonlinear, while Banerjee [Linear Algebra and its Applications, 614 (2021) 82-110] proposed a novel adjacency matrix $A(H)$ for general hypergraphs, whose matrix-based representation preserves typical graph-like spectral features, including the correspondence between row sums and vertex degrees. For the existence of spanning $k$-trees in graphs, authors typically use quotient matrices as tools to achieve expected results. Due to the complex structure of the entropy matrix in hypergraphs. In this paper, by means of a different approach, we establish a relationship between adjacency spectral condition on $A(H)$ and the existence of spanning $k$-hypertree for hypergraph $H$ and verify a sharp lower bound on adjacency spectral radius to guarantee the existence of spanning $k$-hypertrees in uniform hypergraphs.