{
  "abstract": "Let $ρ(G)$ denote the signless Laplacian spectral radius of a graph $G$, and let $\\mathcal C_{n,m}$ be the family of connected graphs of order $n$ and size $m$. A graph is $Q$-maximal if it maximizes $ρ(G)$ over this family. Motivated by a question of Chang and Tam (2011), we study this extremal problem using scalar identities and coronal methods for threshold graphs. From the Perron equations, we derive the identity $S_G(λ)=T_{a,b}(λ)$ at $λ=ρ(G)$, where $a$ and $b$ are the total numbers of zero-vertices and one-vertices. This yields a one-point comparison principle: for a connected threshold graph $H$ with the same block totals, $S_H(z)>T_{a,b}(z)$ at some $z\\ge n$ implies $ρ(H)>z$, allowing spectral comparison without estimating individual Perron coordinates. We prove that every $Q$-maximal threshold graph has character at most three, so its vertex-sequence contains at most three zero--one block pairs. We also show that the quasi-star graph is $Q$-maximal whenever $n\\ge5$ and $n-1\\le m<(n^2+10n-8)/8$. For certain dense parameter pairs, we construct connected threshold graphs with signless Laplacian spectral radii strictly larger than those of both the quasi-star and the connected quasi-complete graph. Thus the corresponding two-candidate assertion for connected graphs fails in general.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07429",
  "author_total": 4,
  "authors": [
    {
      "name": "Di-Jian Wang"
    },
    {
      "name": "Hong-Yan Ge"
    },
    {
      "name": "Shi-Cai Gong"
    },
    {
      "name": "Bo-Jun Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/81pfmhzatdgan.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07429",
  "fulltext_url": null,
  "id": "81pfmhzatdgan",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07429",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07429",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/81pfmhzatdgan.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07429",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07429"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connected graphs having maximum signless Laplacian spectral radius",
  "title_text": "Connected graphs having maximum signless Laplacian spectral radius",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/81pfmhzatdgan",
  "venue": "arXiv",
  "versions": []
}