{
  "id": "2308.07543",
  "version": "v1",
  "published": "2023-08-15T03:18:04.000Z",
  "updated": "2023-08-15T03:18:04.000Z",
  "title": "The $α$-index of graphs without intersecting triangles/quadrangles as a minor",
  "authors": [
    "Yanting Zhang",
    "Ligong Wang"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $A_{\\alpha}$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\\alpha}(G) = \\alpha D(G) + (1 - \\alpha)A(G)$, where $0\\leq\\alpha \\leq1$. The $\\alpha$-index of $G$ is the largest eigenvalue of $A_\\alpha(G)$. Particularly, the matrix $A_0(G)$ (resp. $2A_{\\frac{1}{2}}(G)$) is exactly the adjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng [arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency spectral radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively. Motivated by the above results of He, Li and Feng, in this paper we characterize the extremal graphs with maximum $\\alpha$-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any $0<\\alpha<1$, respectively. As by-products, we determine the extremal graphs with maximum signless Laplacian radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-15T03:18:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficiently large order",
      "intersecting triangles/quadrangles",
      "extremal graphs",
      "adjacency matrix",
      "maximum adjacency spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}