$W_5$-free 图在奇数边数下的谱极值图

中文摘要 AI 辅助

对于固定整数 $k\ge 2$，令 $W_{2k+1}=K_1\vee C_{2k}$ 为一个奇轮图。固定边数的谱极值问题旨在确定 \\[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\\ G \text{ 是 } W_{2k+1}\text{-free}\}, \\] 其中 $\rho(G)$ 表示邻接谱半径。基于此问题，Yu、Li 和 Peng [12] 提出了以下猜想：当 $m-\binom{k}{2}$ 能被 $k$ 整除且 $m$ 足够大时，每个大小为 $m$ 的 $W_{2k+1}$-free 图满足 \\( \rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2} \\)，且等号恰好由 $K_k\vee qK_1$ 取得。对于非零余数类，Yu、Zhang 和 Zhang [13] 提出了以下猜想：设 $r$ 是 $m-\binom{k}{2}$ 除以 $k$ 的非零余数，且 $m$ 足够大。则在大小为 $m$ 的 $W_{2k+1}$-free 图中，$S_{k,m}$ 是唯一具有最大谱半径的图，其中 $S_{k,m}$ 由 $K_k\vee qK_1$ 添加一个顶点 $z$ 并恰好将其连接到 $K_k$ 的 $r$ 个顶点得到。最近，Fang、Zhai 和 Zhang [4] 对 $k\ge 2$ 证实了 Yu--Li--Peng 猜想。Chen、Gao 和 Li [2] 对 $k\ge 3$ 证实了 Yu-Zhang-Zhang 猜想。当 $k=2$ 时，$W_5=K_1\vee C_4$。对于大的奇数 $m$，确定 $\operatorname{spex}(m,W_5)$ 仍然是开放的。在本文中，我们解决了奇数边数问题。我们的结果否定了 $k=2$ 时的 Yu-Zhang-Zhang 猜想。在我们的证明中，一个通用的缺陷界表明只有 $O(1)$ 条边可以位于稠密核心之外。Perron 局部化将其减少到至多一条边。一个离散缺陷不等式迫使完全二部交叉并量化两个匹配缺陷。精确的商多项式比较消除了剩余的交叉边和奇-奇候选。

英文摘要

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.

发表机构

• School of Mathematics and Statistics, and Hubei Key Lab–Math. Sci.,Central China Normal University（华中师范大学数学与统计学院，湖北省数学科学重点实验室）
• School of Mathematics and Big Data,Anhui University of Science and Technology（安徽理工大学数学与大数据学院）

机构由 AI 辅助整理，请以论文原文为准。

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