{
  "id": "2412.11580",
  "version": "v2",
  "published": "2024-12-16T09:13:58.000Z",
  "updated": "2024-12-20T06:03:49.000Z",
  "title": "The existence of a $\\{P_{2},C_{3},P_{5},\\mathcal{T}(3)\\}$-factor based on the size or the $A_α$-spectral radius of graphs",
  "authors": [
    "Xianglong Zhang",
    "Lihua You"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a connected graph of order $n$. A $\\{P_{2},C_{3},P_{5},\\mathcal{T}(3)\\}$-factor of $G$ is a spanning subgraph of $G$ such that each component is isomorphic to a member in $\\{P_{2},C_{3},P_{5},\\mathcal{T}(3)\\}$, where $\\mathcal{T}(3)$ is a $\\{1,2,3\\}$-tree. The $A_{\\alpha}$-spectral radius of $G$ is denoted by $\\rho_{\\alpha}(G)$. In this paper, we obtain a lower bound on the size or the $A_{\\alpha}$-spectral radius for $\\alpha\\in[0,1)$ of $G$ to guarantee that $G$ has a $\\{P_{2},C_{3},P_{5},\\mathcal{T}(3)\\}$-factor, and construct an extremal graph to show that the bound on $A_{\\alpha}$-spectral radius is optimal.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-12-20T06:03:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C35"
    ],
    "keywords": [
      "spectral radius",
      "lower bound",
      "extremal graph",
      "connected graph",
      "spanning subgraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}