The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs

Xiying Yuan, Liqun Qi, Jiayu Shao

Research output: Journal article publicationJournal articleAcademic researchpeer-review

13 Citations (Scopus)

Abstract

Let A(G),L(G) and Q(G) be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph G, respectively. Denote by λ(T) the largest H-eigenvalue of tensor T. Let H be a uniform hypergraph, andH′be obtained from H by inserting a new vertex with degree one in each edge. We prove that λ(Q(H′)) (Q(H)). Denote by ;bsupesup the kth power hypergraph of an ordinary graph G with maximum degree Δ>2. We prove that {λ(Q(;bsupesup))} is a strictly decreasing sequence, which implies Conjecture 4.1 of Hu, Qi and Shao in [4]. We also prove that λ(Q(;bsupesup)) converges to Δ when k goes to infinity. The definition of kth power hypergraph ;bsupesup has been generalized as ;bsupesup We also prove some eigenvalues properties about A(;bsupesup), which generalize some known results. Some related results about L(G) are also mentioned.
Original languageEnglish
Pages (from-to)18-30
Number of pages13
JournalLinear Algebra and Its Applications
Volume490
DOIs
Publication statusPublished - 1 Feb 2016

Keywords

  • 05C50
  • MSC 15A42

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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