Title: A spectral condition for the existence of $C_{2k+1}$ in non-bipartite graphs.
Abstract: A graph $G$ is $H$-free, if it contains no subgraph isomorphic to $H$. 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 $n$? Guo, Lin and Zhao proved a Brualdi-Solheid-Tur\'{a}n type problem on non-bipartite graphs for pentagon. In this paper, we study that if $G$ is a non-bipartite graph with sufficiently large order $n$ and $\rho(G)\geq \rho(K_{\lceil\frac{n-2}{2}\rceil, \lfloor\frac{n-2}{2}\rfloor}\bullet K_3)$, then $G$ contains a $C_{2k+1}$ unless $G\cong K_{\lceil\frac{n-2}{2}\rceil, \lfloor\frac{n-2}{2}\rfloor}\bullet K_3$, where $K_{a, b}\bullet K_3$ is the graph obtained by identifying a vertex of $K_{a,b}$ belonging to the part of size $b$ and a vertex of $K_3$.
Publication Year: 2021
Publication Date: 2021-10-24
Language: en
Type: preprint
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