{
  "abstract": "We study the change in the trace norm and the centred energy of the $A_α$-matrix of a graph under vertex deletion. A pinching argument proves that the trace norm is nondecreasing under passage from an induced subgraph to the original graph, and gives a lower bound in terms of the edges induced by the deleted vertices. Exact perturbation spectra yield sharp upper bounds for deleting a vertex or an independent set of vertices with a common neighbourhood. For $0<α<1/2$, equality in these upper bounds is characterized by star and complete bipartite components, respectively. For the centred energy, we derive explicit deletion bounds and classify all connected equality graphs for $0\\leα<1$: the extremal graphs are stars and complete bipartite graphs for single-vertex and independent-twin deletion, respectively. We also characterize equality in the general-set trace-norm lower bound and prove strictness of the total centred-energy inequality for connected nonregular graphs when $0<α<1$. Examples show that centred local energy can be negative and that the degree deviation term cannot be omitted in general.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24517",
  "author_total": 2,
  "authors": [
    {
      "name": "Ravinder Kumar"
    },
    {
      "name": "Piyush Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dee412qt816f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24517",
  "fulltext_url": null,
  "id": "5dee412qt816f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24517",
  "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": "W7223009397",
  "pdf_url": "https://arxiv.org/pdf/2610.24517",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dee412qt816f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24517",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24517"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Local energy and vertex deletion for $A_α$-matrices",
  "title_text": "Local energy and vertex deletion for A_α-matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/5dee412qt816f",
  "venue": "arXiv",
  "versions": []
}