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Signed Complete Graphs with Maximum Index

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.7151/dmgt.2276
  3. Publisher: Sciendo , 2020
  4. Abstract:
  5. Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ E(G) → {-, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has-1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if is a signed complete graph of order n with k negative edges, k-lt-n-1 and has maximum index, then negative edges form K1 ,k. In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph. We show that the eigenvalues of Γ satisfy the following inequalities:-5 ≤ λn ≤ ≤ λ2 ≤ 3. © 2020 Saieed Akbari et al., published by Sciendo
  6. Keywords:
  7. Complete graph ; Index ; Signed graph
  8. Source: Discussiones Mathematicae - Graph Theory ; Volume 40, Issue 2 , 2020 , Pages 393-403
  9. URL: https://content.sciendo.com/downloadpdf/journals/dmgt/40/2/article-p393.pdf