{
  "id": "2410.07721",
  "version": "v1",
  "published": "2024-10-10T08:41:31.000Z",
  "updated": "2024-10-10T08:41:31.000Z",
  "title": "The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size",
  "authors": [
    "Jing Gao",
    "Xueliang Li"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $\\theta_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\\leq l_2\\leq l_3$ and $l_2\\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $\\theta_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\\geq p\\geq3$ and $p+q\\geq2k+1\\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $\\theta_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $\\theta_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\\mathcal{G}(m,\\theta_{1,p,q})$ which have the largest spectral radius have been determined, where $q\\geq p\\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-10T08:41:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C50"
    ],
    "keywords": [
      "maximum spectral radius",
      "free graph",
      "largest spectral radius",
      "theta graph",
      "distinct vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}