{
  "abstract": "The second Zagreb matrix of a graph $G$ is $Z(G)=D(G)A(G)D(G)$, where $D(G)$ and $A(G)$ are the degree matrix and the adjacency matrix of $G$. For many degree-based weighted adjacency matrices, the trees, unicyclic graphs and bicyclic graphs with the largest spectral radius have a vertex adjacent to all others, and Mei et al. conjectured this for every such matrix on unicyclic graphs. Jiang et al. showed that $Z(G)$ is a counterexample. We prove that, for $q\\in\\{0,1,2\\}$ and every $n\\ge4$, the connected graph of order $n$ and cyclomatic number $q$ maximizing the spectral radius of $Z(G)$ is unique, and is obtained from two adjacent vertices with $q$ common neighbours by attaching $n-q-2$ pendent vertices to them as evenly as possible. The only exceptions are the star for $q=0$ and $n\\le7$, and the triangle with one pendent vertex at each vertex for $q=1$ and $n=6$. This confirms a conjecture of Jiang et al.\\ and all three parts of a conjecture of Shen and Shan for this matrix, with best possible thresholds on $n$. For $n\\ge8$, the maximum value is $\\lfloor (n+q)^2/4\\rfloor+(n+q)/2+7q-2+O(1/n)$, whereas the star gives only $(n-1)^{3/2}$. A key tool is a determinant criterion that reduces spectral comparisons to sign conditions on explicit polynomials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05057",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Wang"
    },
    {
      "name": "Genjie Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17becxqfffngv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05057",
  "fulltext_url": null,
  "id": "17becxqfffngv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05057",
  "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/2611.05057",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17becxqfffngv.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05057",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05057"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs",
  "title_text": "The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/17becxqfffngv",
  "venue": "arXiv",
  "versions": []
}