{
  "abstract": "A graph is $K_{r+1}$-saturated if it is $K_{r+1}$-free and adding any missing edge creates a copy of $K_{r+1}$. Kim, Kim, Kostochka, and O conjectured that $K_{r-1}\\vee(n-r+1)K_1$ minimizes the spectral radius among all $n$-vertex $K_{r+1}$-saturated graphs. They proved the case $r=2$, and the cases $r=3$ and $r\\in\\{4,5\\}$ were subsequently established by Kim, Kostochka, O, Shi, and Wang, and by Wang and Hou, respectively. We settle the conjecture for all $r\\ge3$: if $n\\ge r+1$ and $G$ is an $n$-vertex $K_{r+1}$-saturated graph, then \\[ ρ(G)\\ge \\frac{r-2+\\sqrt{(r-2)^2+4(r-1)(n-r+1)}}{2}, \\] with equality if and only if $G\\cong K_{r-1}\\vee(n-r+1)K_1$. We also prove O's local two-walk conjecture for every $r\\ge2$: \\[ \\sum_{w\\in N_G(v)}d_G(w) \\ge (r-2)d_G(v)+(r-1)(n-r+1) \\qquad(v\\in V(G)). \\] If $G$ has no universal vertex, the inequality holds with the additional term $(r-1)(r-2)$ on the right-hand side. This constant is best possible uniformly in $n$ for every fixed $r$, and gives a strict improvement when $r\\ge3$. A corresponding spectral bound follows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12612",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhengbo Chen"
    },
    {
      "name": "Xiao-Dong Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5m1ty3qv55jyt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12612",
  "fulltext_url": null,
  "id": "5m1ty3qv55jyt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12612",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7212769439",
  "pdf_url": "https://arxiv.org/pdf/2609.12612",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5m1ty3qv55jyt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12612",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12612"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The saturated spectral radius for complete graphs",
  "title_text": "The saturated spectral radius for complete graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/5m1ty3qv55jyt",
  "venue": "arXiv",
  "versions": []
}