{
  "id": "2309.08184",
  "version": "v1",
  "published": "2023-09-15T06:19:49.000Z",
  "updated": "2023-09-15T06:19:49.000Z",
  "title": "On the first two eigenvalues of regular graphs",
  "authors": [
    "Shengtong Zhang"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CO",
    "math.SP"
  ],
  "abstract": "Let $G$ be a regular graph with $m$ edges, and let $\\mu_1, \\mu_2$ denote the two largest eigenvalues of $A_G$, the adjacency matrix of $G$. We show that $$\\mu_1^2 + \\mu_2^2 \\leq \\frac{2(\\omega - 1)}{\\omega} m$$ where $\\omega$ is the clique number of $G$. This confirms a conjecture of Bollob\\'{a}s and Nikiforov for regular graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-15T06:19:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular graph",
      "largest eigenvalues",
      "adjacency matrix",
      "clique number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}