A spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs

Abstract: A graph $G$ is called $H$-free, if it does not contain $H$ as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Tur\'{a}n type problem: what is the maximum spectral radius of an $H$-free graph of order $n$? In this paper, we consider the Brualdi-Solheid-Tur\'{a}n type problem for non-bipartite graphs. Let $K_{a, b}\bullet K_3$ denote the graph obtained by identifying a vertex of $K_{a,b}$ in the part of size $b$ and a vertex of $K_3$. We prove that if $G$ is a non-bipartite graph of order $n$ satisfying $\rho(G)\geq \rho(K_{\lceil\frac{n-2}{2}\rceil, \lfloor\frac{n-2}{2}\rfloor}\bullet K_3)$, then $G$ contains all odd cycles $C_{2l+1}$ for each integer $l\in[2,k]$ unless $G\cong K_{\lceil\frac{n-2}{2}\rceil, \lfloor\frac{n-2}{2}\rfloor}\bullet K_3$, provided that $n$ is sufficiently large with respect to $k$. This resolves the problem posed by Guo, Lin and Zhao (2021).