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

Abstract

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 \[ \operatorname{spex}(m,W_{2k+1}):=\max\{ρ(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho (G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [18] proposed the following conjecture: For large $m$, every $W_{2k+1}$-free graph of size $m$ satisfies $\rho (G{)}^2-(k-1)\rho (G)\le m-(\genfrac{}{}{0ex}{}{k}{2})$ with equality precisely for $K_k\vee q{K}_1$ and $m-\left(\genfrac{}{}{0ex}{}{k}{2}\right)=kq$. Very recently, Fang, Zhai and Zhang [7] confirmed the Yu--Li--Peng conjecture. When $m$ is large, $k\ge 2$ and $m-\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ is not divisible by $k$, the exact solution for the above problem is still open. Regarding this problem, Yu, Zhang, and Zhang [19] proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ 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 $S_{k,m}$ 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 $k\ge 3$.

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