Spectral extremal problems for non-bipartite graphs without odd cycles

TL;DR

This paper determines the maximum spectral radius of non-bipartite graphs avoiding certain short odd cycles, extending previous results and introducing new spectral stability techniques.

Contribution

It establishes the spectral extremal graph for non-bipartite graphs without specific odd cycles, removing size constraints and employing novel proof methods.

Findings

Identifies the extremal graph for spectral radius under cycle restrictions

Extends prior results by removing size constraints

Introduces a new spectral stability proof technique

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, Erd\H{o}s 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 $\{C_3,\ldots ,C_{2\ell -1},C_{2k+1}\}$-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.