{
  "abstract": "For any fixed $k$, we prove a lower bound on the $k$th largest modulus of an eigenvalue of the non-backtracking matrix $B$. Specifically, consider any deterministic or random family of graphs that converges locally to the unimodular Galton-Watson tree with root degree distribution $D$, and set $κ:=\\mathbb E[D(D-1)]/\\mathbb E[D]$. Given $κ>1$ and an exponential-moment bound on the empirical degree distributions, we show that $|λ_k(B)|\\geq\\sqrtκ-o_N(1)$, where $N$ is the number of vertices. When restricted to locally tree-like regular graphs, this recovers a well-known consequence of the Ihara-Bass formula. In the specific case where the graph is generated through the Erdős-Rényi model with expected degree $d>1$, this proves a conjecture of Bordenave, Lelarge, and Massoulié. To do this, we show that the normalized log-determinant of the Bethe-Hessian of the graph is bounded by that of the Bethe-Hessian of its local limit. This bound is violated if the eigenvalues of the non-backtracking matrix are too small. We establish this using an effective-conductance interpretation of the tree Green's function recursion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17529",
  "author_total": 1,
  "authors": [
    {
      "name": "Theo McKenzie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkcrq06ew9031.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17529",
  "fulltext_url": null,
  "id": "dkcrq06ew9031",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17529",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkcrq06ew9031.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "An Alon-Boppana Bound for the Non-Backtracking Operator",
  "title_text": "An Alon-Boppana Bound for the Non-Backtracking Operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/dkcrq06ew9031",
  "venue": "arXiv",
  "versions": []
}