{
  "abstract": "Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs $\\mathcal{F}$, a graph is called $\\mathcal{F}$-free if it contains no member of $\\mathcal{F}$ as a subgraph. The theta graph $θ(l_1,\\ldots,l_k)$ consists of $k$ internally disjoint paths of lengths $l_1,\\ldots,l_k$ with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all $\\{C_3,C_4\\}$-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all $\\{θ(1,2,2),θ(1,2,3)\\}$-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17154",
  "author_total": 2,
  "authors": [
    {
      "name": "Mainak Basunia"
    },
    {
      "name": "Pratima Panigrahi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3525ckfdzabeq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17154",
  "fulltext_url": null,
  "id": "3525ckfdzabeq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17154",
  "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/2609.17154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3525ckfdzabeq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17154"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs",
  "title_text": "Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/3525ckfdzabeq",
  "venue": "arXiv",
  "versions": []
}