{
  "abstract": "Let $G$ be a simple graph on $N$ nodes and $M$ edges, with minimum degree at least $2$. We prove that the non-backtracking Laplacian of $G$ has at least $2M-2N+2c+2b$ real eigenvalues, counted with algebraic multiplicity, where $c$ is the number of connected components of $G$ and $b$ is the number of bipartite ones. More precisely, at least $M-N+2c$ of them lie in $[0,1)$ and at least $M-N+2b$ lie in $(1,2]$. We show that the bound $2M-2N+2c+2b$ is sharp, and that a connected $d$-regular graph attains it if and only if it is strictly Ramanujan, i.e., every adjacency eigenvalue $μ\\ne\\pm d$ satisfies $|μ|<2\\sqrt{d-1}$. For completeness, we also show that the non-backtracking Laplacian has at most $2M-2c-2b$ real eigenvalues.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02194",
  "author_total": 2,
  "authors": [
    {
      "name": "Raffaella Mulas"
    },
    {
      "name": "Leo Torres"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dw17jm91rn5x8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02194",
  "fulltext_url": null,
  "id": "dw17jm91rn5x8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02194",
  "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/2611.02194",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dw17jm91rn5x8.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02194",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02194"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The real eigenvalues of the non-backtracking Laplacian",
  "title_text": "The real eigenvalues of the non-backtracking Laplacian",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/dw17jm91rn5x8",
  "venue": "arXiv",
  "versions": []
}