{
  "abstract": "Let $G$ be a connected graph on at least two vertices, and let $δ(i)$ be the sum of the degrees of the neighbours of a vertex $i$. The \\emph{neighbourhood second Zagreb matrix} $\\NF(G)$ has $(i,j)$-entry $δ(i)δ(j)$ if $i\\sim j$ and $0$ otherwise; its spectral radius is denoted by $\\rNF(G)$. We prove that $\\rNF(G)\\geρ(G)^{5}$, where $ρ(G)$ is the adjacency spectral radius, with equality if and only if $G$ is regular or bipartite semiregular, and we use this to show that the same graphs are exactly the extremal graphs of a Berman--Zhang--type upper bound. We sharpen the estimates $\\demin^{2}ρ(G)\\le\\rNF(G)\\le\\demax^{2}ρ(G)$ by means of the extreme edge weights of $\\NF(G)$, give sharp upper bounds in terms of the maximum degree $\\dmax$, the maximum neighbourhood degree sum $\\demax$, the size $m$ and the clique number $ω(G)$, and determine the extremal graphs for bipartite graphs and trees. For every connected unicyclic graph $G$ of order $n\\ge3$ we prove $32=\\rNF(C_n)\\le\\rNF(G)\\le\\rNF(\\Sp n)$, where $\\Sp n$ is the triangle with $n-3$ pendant edges at one vertex, with equality only for $C_n$ and only for $\\Sp n$, respectively. Finally, we give bounds in terms of the first Zagreb index and for the neighbourhood second Zagreb energy; for lower energy bounds, the extremal graphs are described completely.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06091",
  "author_total": 2,
  "authors": [
    {
      "name": "Subhendu Chakraborty"
    },
    {
      "name": "Priti Prasanna Mondal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zh2p569hq95y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06091",
  "fulltext_url": null,
  "id": "7zh2p569hq95y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06091",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06091",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zh2p569hq95y.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06091",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06091"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Neighbourhood second Zagreb matrix of graphs: spectral radius bounds and extremal graphs",
  "title_text": "Neighbourhood second Zagreb matrix of graphs: spectral radius bounds and extremal graphs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/7zh2p569hq95y",
  "venue": "arXiv",
  "versions": []
}