Spectral extremal problems for non-bipartite graphs without odd cycles

Abstract

A well-known result of Mantel asserts that every n-vertex triangle-free graph G has at most $\lfloor n^2/4\rfloor$ edges. Moreover, Erdős proved that if G is further non-bipartite, then $e\left(G\right)\le \lfloor {(n-1)}^2/4\rfloor +1$. Recently, Lin, Ning and Wu (2021) [35] established a spectral version by showing that if G is a triangle-free non-bipartite graph on n vertices, then $\lambda \left(G\right)\le \lambda \left(S_1\right(T_{n-1,2}\left)\right)$, with equality if and only if $G=S_1\left(T_{n-1,2}\right)$, where $S_1\left(T_{n-1,2}\right)$ is obtained from $T_{n-1,2}$ by subdividing an edge. In this paper, we investigate the maximum spectral radius of a non-bipartite graph without some short odd cycles. Let $C_{2\ell +1}\left(T_{n-2\ell ,2}\right)$ be the graph obtained by identifying a vertex of $C_{2\ell +1}$ and a vertex of the smaller partite set of $T_{n-2\ell ,2}$. We prove that for $1\le \ell <k$ and $n\ge 187k$, if G is an n-vertex $\{C_3,\dots ,C_{2\ell -1},C_{2k+1}\}$-free non-bipartite graph, then $\lambda \left(G\right)\le \lambda \left(C_{2\ell +1}\right(T_{n-2\ell ,2}\left)\right)$, with equality if and only if $G=C_{2\ell +1}\left(T_{n-2\ell ,2}\right)$. This result could be viewed as a spectral analogue of a min-degree result due to Yuan and Peng (2025) [54]. Moreover, our result extends a result of Guo, Lin and Zhao (2021) [20] as well as a recent result of Zhang and Zhao (2023) [59] since we can get rid of the condition that n is sufficiently large. The argument in our proof is quite different and makes use of the classical spectral stability method and the double-eigenvector technique. The main innovation lies in a more clever argument that guarantees a subgraph to be bipartite after removing few vertices, which may be of independent interest.