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A bound for the p-domination number of a graph in terms of its eigenvalue multiplicities

Abiad, A ; Sharif University of Technology | 2023

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2022.11.008
  3. Publisher: Elsevier Inc , 2023
  4. Abstract:
  5. Let G be a connected graph of order n with domination number γ(G). Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue λ of G with multiplicity mG(λ), it holds that γ(G)≤n−mG(λ). Using techniques from the theory of star sets, in this work we prove that the same bound holds when λ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the p-domination number, and we apply it to extend the aforementioned spectral bound to the p-domination number using the adjacency and Laplacian eigenvalue multiplicities. © 2022 The Authors
  6. Keywords:
  7. Adjacency matrix ; Eigenvalue multiplicity ; Laplacian matrix ; p-domination number ; Rank ; Total domination number
  8. Source: Linear Algebra and Its Applications ; Volume 658 , 2023 , Pages 319-330 ; 00243795 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0024379522004086