The main eigenvalues of signed graphs

Akbari, S ; Sharif University of Technology | 2021

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2020.04.029
  3. Publisher: Elsevier Inc , 2021
  4. Abstract:
  5. A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc
  6. Keywords:
  7. Eigenvalues and eigenfunctions ; Graph structures ; Graphic methods ; Adjacency matrices ; Cayley graphs ; Distance regular graph ; Edge-transitive graphs ; Main eigenvalue ; Non-orthogonal ; Ordered pairs ; Signed graphs ; Graph theory
  8. Source: Linear Algebra and Its Applications ; Volume 614 , 2021 , Pages 270-280 ; 00243795 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0024379520302238