The extremal spectral radii of \(k\) -uniform supertrees

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• 作者：Honghai Li ; Jia-Yu Shao ; Liqun Qi
• 关键词：Hypergraph ; Spectral radius ; Adjacency tensor ; Signless Laplacian tensor ; Incidence \(Q\) ; tensor ; Supertree
• 刊名：Journal of Combinatorial Optimization
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：32
• 期：3
• 页码：741-764
• 全文大小：538 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Mathematical Modeling and IndustrialMathematics
  Theory of Computation
  Optimization
  Operation Research and Decision Theory
  
• 出版者：Springer Netherlands
• ISSN：1573-2886
• 卷排序：32

文摘

In this paper, we study some extremal problems of three kinds of spectral radii of \(k\)-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidence \(Q\)-spectral radius). We call a connected and acyclic \(k\)-uniform hypergraph a supertree. We introduce the operation of “moving edges” for hypergraphs, together with the two special cases of this operation: the edge-releasing operation and the total grafting operation. By studying the perturbation of these kinds of spectral radii of hypergraphs under these operations, we prove that for all these three kinds of spectral radii, the hyperstar \(\mathcal {S}_{n,k}\) attains uniquely the maximum spectral radius among all \(k\)-uniform supertrees on \(n\) vertices. We also determine the unique \(k\)-uniform supertree on \(n\) vertices with the second largest spectral radius (for these three kinds of spectral radii). We also prove that for all these three kinds of spectral radii, the loose path \(\mathcal {P}_{n,k}\) attains uniquely the minimum spectral radius among all \(k\)-th power hypertrees of \(n\) vertices. Some bounds on the incidence \(Q\)-spectral radius are given. The relation between the incidence \(Q\)-spectral radius and the spectral radius of the matrix product of the incidence matrix and its transpose is discussed.