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

TL;DR

This paper establishes a spectral condition involving the spectral radius of a graph that guarantees the presence of all odd cycles up to a certain length in large non-bipartite graphs, resolving a problem posed in 2021.

Contribution

It provides a spectral criterion for the existence of all odd cycles up to a certain length in large non-bipartite graphs, extending previous work and solving an open problem.

Findings

Spectral radius threshold ensures all small odd cycles are present.

Characterization of extremal graphs achieving the spectral bound.

Resolution of a conjecture by Guo, Lin, and Zhao (2021).

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).