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Some results on the Laplacian spread conjecture

Afshari, B ; Sharif University of Technology | 2019

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2019.03.003
  3. Publisher: Elsevier Inc , 2019
  4. Abstract:
  5. For a graph G of order n, let λ 2 (G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ 2 (G)+λ 2 (G‾)≥1, where G‾ is the complement of G. For any x∈R n , let ∇ x ∈R (n2) be the vector whose {i,j}-th entry is |x i −x j |. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈R n with zero mean satisfy ‖∇ f −∇ g ‖ 2 ≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs. © 2019 Elsevier Inc
  6. Keywords:
  7. Laplacian eigenvalues of graphs ; Laplacian spread ; Eigenvalues and eigenfunctions ; Graph G ; Laplacian eigenvalues ; Laplacians ; Orthonormal vectors ; Permutation graph ; Laplace transforms
  8. Source: Linear Algebra and Its Applications ; Volume 574 , 2019 , Pages 22-29 ; 00243795 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0024379519300977