{
  "id": "2302.01916",
  "version": "v1",
  "published": "2022-11-02T06:38:38.000Z",
  "updated": "2022-11-02T06:38:38.000Z",
  "title": "Spectral radius of graphs of given size with forbidden subgraphs",
  "authors": [
    "Yuxiang Liu",
    "Ligong Wang"
  ],
  "comment": "16 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\rho(G)$ be the spectral radius of a graph $G$ with $m$ edges. Let $S_{m-k+1}^{k}$ be the graph obtained from $K_{1,m-k}$ by adding $k$ disjoint edges within its independent set. Nosal's theorem states that if $\\rho(G)>\\sqrt{m}$, then $G$ contains a triangle. Zhai and Shu showed that any non-bipartite graph $G$ with $m\\geq26$ and $\\rho(G)\\geq\\rho(S_{m}^{1})>\\sqrt{m-1}$ contains a quadrilateral unless $G\\cong S_{m}^{1}$ [M.Q. Zhai, J.L. Shu, Discrete Math. 345 (2022) 112630]. Wang proved that if $\\rho(G)\\geq\\sqrt{m-1}$ for a graph $G$ with size $m\\geq27$, then $G$ contains a quadrilateral unless $G$ is one of four exceptional graphs [Z.W. Wang, Discrete Math. 345 (2022) 112973]. In this paper, we show that any non-bipartite graph $G$ with size $m\\geq51$ and $\\rho(G)\\geq\\rho(S_{m-1}^{2})>\\sqrt{m-2}$ contains a quadrilateral unless $G$ is one of three exceptional graphs. Moreover, we show that if $\\rho(G)\\geq\\rho(S_{\\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\\geq74$, then $G$ contains a $C_{5}^{+}$ unless $G\\cong S_{\\frac{m+4}{2},2}^{-}$, where $C_{t}^{+}$ denotes the graph obtained from $C_{t}$ and $C_{3}$ by identifying an edge, $S_{n,k}$ denotes the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denotes the graph obtained by deleting an edge incident to a vertex of degree two, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-02T06:38:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C35"
    ],
    "keywords": [
      "spectral radius",
      "forbidden subgraphs",
      "non-bipartite graph",
      "exceptional graphs",
      "discrete math"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}