Spectral Radius of Graphs with Given Size and Odd Girth

• Published in: The Electronic journal of combinatorics Vol. 31; no. 1
• Main Authors: Lou, Zhenzhen, Lu, Lu, Huang, Xueyi
• Format: Journal Article
• Language: English
• Published: 08.03.2024
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/11720

Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $\eta(m,k)>\sqrt{m-k+3}$ such that every non bipartite graph $G$ with size $m$ and spectral radius $\rho\ge \eta(m,k)$ must contain an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of the complete bipartite graph $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of Zhai and Shu [Discrete Math. 345 (2022)] and settles a conjecture of Li and Peng  [The Electronic J. Combin. 29 (4) (2022)] as well.