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Some criteria for a signed graph to have full rank

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2020.111910
  3. Publisher: Elsevier B.V , 2020
  4. Abstract:
  5. A weighted graph Gω consists of a simple graph G with a weight ω, which is a mapping, ω: E(G)→Z∖{0}. A signed graph is a graph whose edges are labelled with −1 or 1. In this paper, we characterize graphs which have a sign such that their signed adjacency matrix has full rank, and graphs which have a weight such that their weighted adjacency matrix does not have full rank. We show that for any arbitrary simple graph G, there is a sign σ so that Gσ has full rank if and only if G has a {1,2}-factor. We also show that for a graph G, there is a weight ω so that Gω does not have full rank if and only if G has at least two {1,2}-factors. © 2020 Elsevier B.V
  6. Keywords:
  7. Rank ; Signed adjacency matrix ; Signed graph ; Weighted adjacency matrix ; Weighted graph
  8. Source: Discrete Mathematics ; Volume 343, Issue 8 , 2020
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0012365X20301023