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Line graphs and nordhaus–gaddum-type bounds for self-loop graphs

Akbari, S ; Sharif University of Technology | 2024

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  1. Type of Document: Article
  2. DOI: 10.1007/s40840-024-01714-3
  3. Publisher: 2024
  4. Abstract:
  5. Let GS be the graph obtained by attaching a self-loop at every vertex in S⊆V(G) of a simple graph G of order n. In this paper, we explore several new results related to the line graph L(GS) of GS. Particularly, we show that every eigenvalue of L(GS) must be at least -2, and relate the characteristic polynomial of the line graph L(G) of G with the characteristic polynomial of the line graph L(G^) of a self-loop graph G^, which is obtained by attaching a self-loop at each vertex of G. Then, we provide some new bounds for the eigenvalues and energy of GS. As one of the consequences, we obtain that the energy of a connected regular complete multipartite graph is not greater than the energy of the corresponding self-loop graph. Lastly, we establish a lower bound of the spectral radius in terms of the first Zagreb index M1(G) and the minimum degree δ(G), as well as proving two Nordhaus–Gaddum-type bounds for the spectral radius and the energy of GS, respectively. © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2024
  6. Keywords:
  7. 05C50 ; 05C90 ; 05C92 ; Adjacency spectrum ; Energy ; Line graph ; Nordhaus–Gaddum-type bounds ; Self-loop graph
  8. Source: Bulletin of the Malaysian Mathematical Sciences Society ; Volume 47, Issue 4 , 2024 ; 01266705 (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s40840-024-01714-3