The edge spectral extremal problem for odd wheels in nonzero residue classes

For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine $$\mathrm{spex}(m,W_{2k+1}):=\max \{\rho (G):e(G)=m,G\text{ is }{W}_{2k+1}\text{-free}\},$$ where $\rho (G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng~\cite{YLP} proposed the following conjecture: When $m-\left(\genfrac{}{}{0px}{}{k}{2}\right)$ is divisible by $k$ and $m$ is large, every $W_{2k+1}$-free graph of size $m$ satisfies $\rho (G{)}^2-(k-1)\rho (G)\le m-\left(\genfrac{}{}{0px}{}{k}{2}\right)$ with equality precisely for $K_k\vee q{K}_1$. Very recently, Fang, Zhai and Zhang~\cite{FangZhaiZhang} confirmed the Yu–Li–Peng conjecture for $k\ge 3$. When $m$ is large, $k\ge 3$ and $m-\left(\genfrac{}{}{0px}{}{k}{2}\right)$ is not divisible by $k$, the above exact problem is still open. Regarding this problem, Yu, Zhang, and Zhang~\cite{YZZ} proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\left(\genfrac{}{}{0px}{}{k}{2}\right)$ is divided by $k$. Then $S_{k,m}$ is the unique graph among $W_{2k+1}$-free graphs of size $m$ having maximum spectral radius, where $S_{m,k}$ is obtained from $K_k\vee q{K}_1$ by adding a vertex $z$ and joining it to exactly $r$ vertices of the $K_k$. In this paper we address this problem in each nonzero residue class. Our result completely settles the Yu-Zhang-Zhang conjecture for $k\ge 3$. The proof is developed by some new techniques and methods such as dense-core reduction, edge-spectral stability, Perron localization, and discrete defect quantization.