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 $e(G)\le \lfloor (n-1{)}^2/4\rfloor +1$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] established a spectral version by showing that if $G$ is a triangle-free non-bipartite graph on $n$ vertices, then $\lambda (G)\le \lambda (S_1(T_{n-1,2}))$, with equality if and only if $G=S_1(T_{n-1,2})$, where $S_1(T_{n-1,2})$ 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 $C_{2\ell +1}(T_{n-2\ell ,2})$ 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 $\lambda (G)\le \lambda (C_{2\ell +1}(T_{n-2\ell ,2}))$, with equality if and only if $G=C_{2\ell +1}(T_{n-2\ell ,2})$. This result could be viewed as a spectral analogue of a min-degree result due to Yuan and Peng [European J. Combin. 127 (2025)]. Moreover, our result extends a result of Guo, Lin and Zhao [Linear Algebra Appl. 627 (2021)] as well as a recent result of Zhang and Zhao [Discrete Math. 346 (2023)] 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.

23 pages. Any suggestions are welcome

References in corpus (9)

• Eigenvalues and triangles in graphs
• Extensions of Erdős-Gallai Theorem and Luo's Theorem with Applications
• Spectral extremal graphs for the bowtie
• The maximum spectral radius of non-bipartite graphs forbidding short odd cycles
• Refinement on spectral Turán's theorem
• Eigenvalues and cycles of consecutive lengths
• A spectral Erdős-Faudree-Rousseau theorem
• Spectral supersaturation: Triangles and bowties
• Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite