{
  "id": "2401.03787",
  "version": "v1",
  "published": "2024-01-08T10:08:15.000Z",
  "updated": "2024-01-08T10:08:15.000Z",
  "title": "On maximum spectral radius of $\\{H(3,3),~H(4,3)\\}$-free graphs",
  "authors": [
    "Amir Rehman",
    "S. Pirzada"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a simple connected graph of size $m$. Let $A$ be the adjacency matrix of $G$ and let $\\rho(G)$ be the spectral radius of $G$. A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. Let $H(\\ell,3)$ be the graph formed by taking a cycle of length $\\ell$ and a triangle on a common vertex. Recently, Li, Lu and Peng [Y. Li, L. Lu, Y. Peng, Spectral extremal graphs for the bowtie, Discrete Math. 346(12) (2023) 113680.] showed that the unique $m$-edge $H(3,3)$-free spectral extremal graph is the join of $K_2$ with an independent set of $\\frac{m-1}{2}$ vertices if $m\\ge 8$ and the condition $m\\ge 8$ is tight. In particular, if $G$ does not contain $H(3,3)$ as induced subgraph, they proved that $\\rho(G) \\leq \\frac{1+\\sqrt{4m-3}}{2} $ and equality holds when $G$ is isomorphic to $S_{\\frac{m+3}{2},2}$. Note that Li et al. denoted $H(3,3)$ by $F_2$. In this paper, we find the maximum spectral radius and identify the graph with the largest spectral radius among all \\{$H(3,3), H(4,3)$\\}-free graphs of size odd $m$, where $m\\geq 259$. Coincidentally, we show that $\\rho(G) \\leq \\frac{1+\\sqrt{4m-3}}{2}$ when $G$ forbids both $H(3,3)$ and $H(4,3)$. In our case, the equality holds when $G$ is isomorphic to the same graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-08T10:08:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C12",
      "15A18"
    ],
    "keywords": [
      "maximum spectral radius",
      "free graphs",
      "free spectral extremal graph",
      "equality holds",
      "largest spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}