2020/11/30 by Saieed Akbari, Sebastian M. Cioabă, Samira Goudarzi +3
Computer Science · Mathematics · #Algebra over a field #Applied mathematics #Calculus (dental) #Combinatorics #Graph theory and applications #Mathematics #Matrix Theory and Algorithms #Pure mathematics #Topological and Geometric Data Analysis #cs.DM #math.CO
paper · pdf · doi:10.1016/j.laa.2021.01.003
openalex publication_date 2021/01/12 · arxiv created 2021/01/21 · arxiv updated 2021/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.