Ordering uniform supertrees by their spectral radii

Abstract

A connected and acyclic hypergraph is called a supertree. In this paper we mainly focus on the spectral radii of uniform supertrees. Li, Shao and Qi determined the first two k-uniform supertrees with large spectral radii among all the k-uniform supertrees on n vertices [H. Li, J. Shao, L. Qi, The extremal spectral radii of k-uniform supertrees, arXiv:1405.7257v1, May 2014]. By applying the operation of moving edges on hypergraphs and using the weighted incidence matrix method we extend the above order to the fourth k-uniform supertree.

The largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

“…In 2008, Lim [11] proposed the study of the spectra of hypergraphs by using the spectra of tensors. Recently, there has been a lot of activity concerning spectral hypergraph theory [8,9,10,13,14,15,23]. Li, Shao and Qi [9] gave the operations of moving edges, edgereleasing and total grafting on hypergraphs, and determined the first two spectral radii of k-uniform supertrees with given n vertices.…”

“…Li, Shao and Qi [9] gave the operations of moving edges, edgereleasing and total grafting on hypergraphs, and determined the first two spectral radii of k-uniform supertrees with given n vertices. Many scholars investigated the extreme spectral radius of k-uniform hypergraphs on other conditions [14,15,23]. Xiao, Wang and Lu [22] investigated the spectral radius of the k-uniform hypergraph when the uniform hypergraph is perturbed by 2-switch operation, and determined the maximum spectral radius of k-uniform supertrees with given a degree sequence.…”

On some properties of the α-spectral radius of the k-uniform hypergraph

“…The spectral radii of k-uniform supertrees have been studied by many authors (see [3,13,14]). In [15,16], Yuan et al determined the top ten supertrees with the maximum spectral radii. And in [4], Guo et al determined the unique non-hyper-caterpillar with maximum adjacency spectral radius among N C(m, d).…”

“…The spectral radii of k-uniform supertrees have been studied by many authors (see [3,13,14]). In [15,16] Note that the supertrees with the first ten largest spectral radii are caterpillar supertrees. It is natural to ask that: (a).…”

The effect on the spectral radius of r-graphs by grafting or contracting edges

“…In 2015, Li, Shao and Qi [7] studied some extremal spectral properties of the classes of uniform supertrees with n vertices, and determined that the hyperstar attains uniquely the maximal spectral radius among all uniform supertrees with given size. Yuan et al [20,19] determined the ordering of supertrees with larger spectral radius. Fan et al [6] determined the hypergraphs with maximum spectral radius over all unicyclic hypergraphs, over linear or power unicyclic hypergraphs with given girth, over linear or power bicyclic hypergrphs, respectively.…”