{
  "id": "1809.07718",
  "version": "v1",
  "published": "2018-09-20T16:13:15.000Z",
  "updated": "2018-09-20T16:13:15.000Z",
  "title": "On the largest $A_α$-spectral radius of cacti",
  "authors": [
    "Shaohui Wang",
    "Chunxiang Wang",
    "Jia-Bao Liu",
    "Bing Wei"
  ],
  "comment": "submitted",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $A(G)$ be the adjacent matrix and $D(G)$ the diagonal matrix of the degrees of a graph $G$, respectively. For $0 \\leq \\alpha \\leq 1$, the $A_{\\alpha}$ matrix $A_{\\alpha}(G) = \\alpha D(G) +(1-\\alpha)A(G)$ is given by Nikiforov. Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\\frac{1}{2}}$ is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The $A_{\\alpha}$-spectral radius of a cactus graph with $n$ vertices and $k$ cycles is explored. The outcomes obtained in this paper can imply previous bounds of Nikiforov et al., and Lov\\'{a}sz and Pelik\\'{a}n. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-20T16:13:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral radius",
      "adjacent matrix",
      "diagonal matrix",
      "signless laplacian matrix",
      "corresponding extremal graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}