A note on the Bollobás-Nikiforov conjecture

Bollobás and Nikiforov [2] proposed a conjecture that for any non-complete graph G with m edges and clique number ω, the following inequality holds:λ12+λ22≤2(1−1ω)m, where λ1 and λ2 are the two largest eigenvalues of the adjacency matrix A(G). Later, Elphick, Linz, and Wocjan [6] proposed a generali...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 710; pp. 230 - 242
Main Authors Zeng, Jiasheng, Zhang, Xiao-Dong
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2025
Subjects
Adjacency matrix
Bollobás-Nikiforov conjecture
Eigenvalues of graph
Line graph
Triangle
Line graph
05C50
Triangle
Adjacency matrix
Bollobás-Nikiforov conjecture
05C35
Eigenvalues of graph
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Summary:Bollobás and Nikiforov [2] proposed a conjecture that for any non-complete graph G with m edges and clique number ω, the following inequality holds:λ12+λ22≤2(1−1ω)m, where λ1 and λ2 are the two largest eigenvalues of the adjacency matrix A(G). Later, Elphick, Linz, and Wocjan [6] proposed a generalization of this conjecture. In this paper, we prove that the conjecture proposed by Bollobás and Nikiforov holds for both line graphs and graphs with at most 627m32 triangles, and that the generalized conjecture holds for both line graphs with additional conditions and graphs with not many triangles, which extends and strengthens some known results.
ISSN:0024-3795
DOI:10.1016/j.laa.2025.01.037