Spectral extremal graphs for $W_5$-free graphs with odd size

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 $ρ(G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [12] proposed the following conjecture: When $m-\binom{k}{2}$ is divisible by $k$ and $m$ is large, every $W_{2k+1}$-free graph of size $m$ satisfies \( ρ(G)^2-(k-1)ρ(G)\le m-\binom{k}{2} \) with equality precisely for $K_k\vee qK_1$. For nonzero residue class, Yu, Zhang, and Zhang [13] proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\binom{k}{2}$ is divided by $k$ and $m$ is large. 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 $K_k\vee qK_1$ by adding a vertex $z$ and joining it to exactly $r$ vertices of the $K_k$. Very recently, Fang, Zhai and Zhang [4] confirmed the Yu--Li--Peng conjecture for $k\ge 2$. Chen, Gao and Li [2] confirmed Yu-Zhang-Zhang conjecture for $k\ge 3$. When $k=2$, then $W_5=K_1\vee C_4$. For large odd $m$, determining $\operatorname{spex}(m,W_5)$ is still open. In this paper we address the odd-size problem. Our result disproved Yu-Zhang-Zhang conjecture for $k= 2$. In our proof, a universal defect bound shows that only $O(1)$ edges can lie outside the dense core. Perron localization then reduces this to at most one edge. A discrete defect inequality forces the complete bipartite crossing and quantizes the two matching deficiencies. Exact quotient-polynomial comparisons eliminate the remaining cross-edge and odd--odd candidates.