{
  "id": "2403.18522",
  "version": "v1",
  "published": "2024-03-27T12:54:47.000Z",
  "updated": "2024-03-27T12:54:47.000Z",
  "title": "On the $A_α$-index of graphs with given order and dissociation number",
  "authors": [
    "Zihan Zhou",
    "Shuchao Li"
  ],
  "comment": "16 pages; 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of $G$. The adjacency matrix and the degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G),$ respectively. In 2017, Nikiforov proposed the $A_\\alpha$-matrix: $A_\\alpha(G)=\\alpha D(G)+(1-\\alpha)A(G),$ where $\\alpha\\in[0,1].$ The largest eigenvalue of this novel matrix is called the $A_\\alpha$-index of $G.$ In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest $A_\\alpha$-index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the $n$-vertex graphs having the minimum $A_\\alpha$-index with dissociation number $\\tau$, where $\\tau\\geqslant\\lceil\\frac{2}{3}n\\rceil.$ Finally, we identify all the connected $n$-vertex graphs with dissociation number $\\tau\\in\\{2,\\lceil\\frac{2}{3}n\\rceil,n-1,n-2\\}$ having the minimum $A_\\alpha$-index.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-27T12:54:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A18"
    ],
    "keywords": [
      "dissociation number",
      "maximum dissociation set",
      "vertex graphs",
      "bipartite graph",
      "degree diagonal matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}