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2024-03-13 08:39:06

This https://arxiv.org/abs/2304.00716 has been replaced.
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A spectral extremal problem on non-bipartite triangle-free graphs
A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $λ(G)\le \sqrt{m}$, where the equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $λ_1^2(G) + λ_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the…