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On a question of Haemers regarding vectors in the nullspace of Seidel matrices

Akbari, S ; Sharif University of Technology | 2021

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2021.01.003
  3. Publisher: Elsevier Inc , 2021
  4. Abstract:
  5. In 2011, Haemers asked the following question: If S is the Seidel matrix of a graph of order n and S is singular, does there exist an eigenvector of S corresponding to 0 which has only ±1 elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number N, there exists a graph whose Seidel matrix S is singular such that for any integer vector in the nullspace of S, the absolute value of any entry in this vector is more than N. We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel matrices. Finally, we obtain some properties of the graphs which affirm the above question. © 2021 Elsevier Inc
  6. Keywords:
  7. Vectors ; Absolute values ; Integer vectors ; Natural number ; Null space ; Matrix algebra
  8. Source: Linear Algebra and Its Applications ; Volume 615 , 2021 , Pages 194-206 ; 00243795 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0024379521000057