{
  "id": "2302.03313",
  "version": "v1",
  "published": "2023-02-07T08:26:38.000Z",
  "updated": "2023-02-07T08:26:38.000Z",
  "title": "Spectral conditions for forbidden subgraphs in bipartite graphs",
  "authors": [
    "Yuan Ren",
    "Jing Zhang",
    "Zhiyuan Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is $H$-free, if it contains no $H$ as a subgraph. A graph is said to be \\emph{$H$-minor free}, if it does not contain $H$ as a minor. In recent years, Nikiforov asked that what is the maximum spectral radius of an $H$-free graph of order $n$? In this paper, we consider about some Brualdi-Solheid-Tur\\'{a}n type problems on bipartite graphs. In 2015, Zhai, Lin and Gong proved that if $G$ is a bipartite graph with order $n \\geq 2k+2$ and $\\rho(G)\\geq \\rho(K_{k,n-k})$, then $G$ contains a $C_{2k+2}$ unless $G \\cong K_{k,n-k}$ [Linear Algebra Appl. 471 (2015)]. Firstly, we give a new and more simple proof for the above theorem. Secondly, we prove that if $G$ is a bipartite graph with order $n \\geq 2k+2$ and $\\rho(G)\\geq \\rho(K_{k,n-k})$, then $G$ contains all $T_{2k+3}$ unless $G \\cong K_{k,n-k}$. Finally, we prove that among all outerplanar bipartite graphs on $n>344569$ vertices, $K_{1,n-1}$ attains the maximum spectral radius.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-07T08:26:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "forbidden subgraphs",
      "spectral conditions",
      "maximum spectral radius",
      "linear algebra appl",
      "outerplanar bipartite graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}