{
  "id": "1805.05808",
  "version": "v1",
  "published": "2018-05-15T14:31:47.000Z",
  "updated": "2018-05-15T14:31:47.000Z",
  "title": "A note on the $A_α$-spectral radius of graphs",
  "authors": [
    "Huiqiu Lin",
    "Xing Huang",
    "Jie Xue"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\\alpha\\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix $A_{\\alpha}(G)$ as $A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\\leq i\\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$. Let $s=0,1$. Nikiforov and Rojo [On the $\\alpha$-index of graphs with pendent paths, Linear Algebra Appl. 550 (2018) 87--104] conjectured that $\\rho_{\\alpha}(G_{p,s,q}(u,v))<\\rho_{\\alpha}(G_{p-1,s,q+1}(u,v))$ if $p\\geq q+2.$ In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal $A_{\\alpha}$-spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal $A_{\\alpha}$-spectral radius with fixed order and matching number. These results generalize some known results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-15T14:31:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral radius",
      "fixed order",
      "linear algebra appl",
      "diagonal matrix",
      "extremal tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}