A note on the Bollobás-Nikiforov conjecture

Bollob & aacute;s and Nikiforov [2] proposed a conjecture that for any non-complete graph C with m edges and clique number w, the following inequality holds: ( ) a2 i + a2 2 <= 2 1 - 1 m, w where ai and a2 are the two largest eigenvalues of the adjacency matrix A (C). Later, Elphick, Linz, and Wocjan [6] proposed a generalization of this conjecture. In this paper, we prove that the conjecture proposed by Bollob & aacute;s and Nikiforov 2 holds for both line graphs and graphs with at most root s 2 m 3 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.