{
  "id": "2210.16480",
  "version": "v1",
  "published": "2022-10-29T03:28:12.000Z",
  "updated": "2022-10-29T03:28:12.000Z",
  "title": "The $A_α$ spectral radius of $k$-connected graphs with given diameter",
  "authors": [
    "Xichan Liu",
    "Ligong Wang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The $A_\\alpha$ matrix of a graph $G$ is defined as the convex combination of its adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, that is, $A_\\alpha(G)=\\alpha A(G)+(1-\\alpha)D(G)$, where $\\alpha\\in[0,1]$, whose largest eigenvalue is called the $A_\\alpha$ spectral radius or $A_\\alpha$-index of $G$. Let $\\mathcal{G}_{n,k}^d$ be the set of $k$-connected graphs of order $n$ with diameter $d$. In this paper, we determine the graphs with maximum $A_\\alpha$ spectral radius among all graphs in $\\mathcal{G}_{n,k}^d$ for any $\\alpha\\in[0,1)$, where $k\\geq2$ and $d\\geq2$, which generalizes the results for adjacency matrix in [P. Huang, W.C. Shiu, P.K. Sun, Linear Algebra Appl., 2016, Theorem 3.6] and signless Laplacian matrix in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl., 2021, Theorem 3.4]. Furthermore, we also obtaine some bounds of the set of graphs $\\mathcal{G}_{n,k}^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-29T03:28:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral radius",
      "connected graphs",
      "linear algebra appl",
      "adjacency matrix",
      "degree diagonal matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}