{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28556",
  "author_total": 3,
  "authors": [
    {
      "name": "Jing Gao"
    },
    {
      "name": "Xianya Geng"
    },
    {
      "name": "Shuchao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/48jbevxe705ys.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28556",
  "fulltext_url": null,
  "id": "48jbevxe705ys",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28556",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7214360594",
  "pdf_url": "https://arxiv.org/pdf/2609.28556",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/48jbevxe705ys.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.28556",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28556"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral extremal graphs for $W_5$-free graphs with odd size",
  "title_text": "Spectral extremal graphs for W_5-free graphs with odd size",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/48jbevxe705ys",
  "venue": "arXiv",
  "versions": []
}