A refinement on spectral Mantel’s theorem

The well-known Mantel’s theorem states that $e\left(G\right)\le \lfloor n^2/4\rfloor$ for every $n$-vertex triangle-free graph $G$. In 1970, Nosal showed a spectral version of Mantel’s theorem, which states that $\rho \left(G\right)\le \sqrt{m}$ for every triangle-free graph $G$ on $m$ edges. Later, Nikiforov proved that the equality holds in Nosal’s bound if and only if $G$ is a complete bipartite graph. Lin, Ning, and Wu [Combin. Probab. Comput. 30 (2021)] first gave a result on non-bipartite triangle-free graphs. They proved that $\rho \left(G\right)\le \sqrt{m-1}$, and equality holds if and only if $G$ is a 5-cycle. Their result is actually stronger than Mantel’s theorem. Recently, Li, Feng and Peng [Electron. J. Combin. 31 (2024)] characterized the extremal non-bipartite triangle-free graphs with maximal spectral radius for even $m$. Furthermore, they proposed a question as follows: what is the extremal graph with maximal spectral radius over all non-bipartite $\{C_3,C_5,\dots ,C_{2\ell +1}\}$-free graphs of even size $m$? This question is completely solved in this paper. Here we use a different technique, and we call it residual function, which can present more concise proofs on related problems. Finally, a general question is proposed for further research.