{
  "id": "2402.09160",
  "version": "v3",
  "published": "2024-02-14T13:24:09.000Z",
  "updated": "2024-07-04T10:10:51.000Z",
  "title": "At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian",
  "authors": [
    "Lies Beers",
    "Raffaella Mulas"
  ],
  "comment": "Added new results in Section 3 (Theorem 3.12 - Proposition 3.16)",
  "categories": [
    "math.CO",
    "math.SP"
  ],
  "abstract": "For a graph with largest normalized Laplacian eigenvalue $\\lambda_N$ and (vertex) coloring number $\\chi$, it is known that $\\lambda_N\\geq \\chi/(\\chi-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the multiplicity of $\\chi/(\\chi-1)$. We then describe a family of graphs with largest eigenvalue $\\chi/(\\chi-1)$. We also study the spectrum of the $1$-sum of two graphs (also known as graph joining or coalescing), with a focus on the maximal eigenvalue. Finally, we give upper bounds on $\\lambda_N$ in terms of $\\chi$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-07-04T10:10:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "largest eigenvalue",
      "chromatic bounds",
      "largest normalized laplacian eigenvalue",
      "upper bounds",
      "maximal eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}