{
  "id": "2105.10903",
  "version": "v1",
  "published": "2021-05-23T10:20:25.000Z",
  "updated": "2021-05-23T10:20:25.000Z",
  "title": "On the $A_α$ spectral radius of strongly connected digraphs",
  "authors": [
    "Weige Xi",
    "Ligong Wang"
  ],
  "comment": "20 pages,6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \\cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al. \\cite{LWCL} extended the definition to digraphs. For any real $\\alpha\\in[0,1]$, the matrix $A_\\alpha(G)$ of a digraph $G$ is defined as $$A_\\alpha(G)=\\alpha D(G)+(1-\\alpha)A(G).$$ The largest modulus of the eigenvalues of $A_\\alpha(G)$ is called the $A_\\alpha$ spectral radius of $G$, denoted by $\\lambda_\\alpha(G)$. This paper proves some extremal results about the spectral radius $\\lambda_\\alpha(G)$ that generalize previous results about $\\lambda_0(G)$ and $\\lambda_{\\frac{1}{2}}(G)$. In particular, we characterize the extremal digraph with the maximum (or minimum) $A_\\alpha$ spectral radius among all $\\widetilde{\\infty}$-digraphs and $\\widetilde{\\theta}$-digraphs on $n$ vertices. Furthermore, we determine the digraphs with the second and the third minimum $A_\\alpha$ spectral radius among all strongly connected bicyclic digraphs. For $0\\leq\\alpha\\leq\\frac{1}{2}$, we also determine the digraphs with the second, the third and the fourth minimum $A_\\alpha$ spectral radius among all strongly connected digraphs on $n$ vertices. Finally, we characterize the digraph with the minimum $A_\\alpha$ spectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-23T10:20:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A18"
    ],
    "keywords": [
      "spectral radius",
      "strongly connected digraphs",
      "diagonal matrix",
      "adjacency matrix",
      "complete bipartite subdigraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}