{
  "id": "2406.07132",
  "version": "v1",
  "published": "2024-06-11T10:28:11.000Z",
  "updated": "2024-06-11T10:28:11.000Z",
  "title": "Spanning trees and signless Laplacian spectral radius in graphs",
  "authors": [
    "Sufang Wang",
    "Wei Zhang"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf degree among all the vertices of $T$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal degree matrix of $G$. Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of $G$. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. In this paper, we investigate the connection between the spanning tree and the signless Laplacian spectral radius of $G$, and put forward a sufficient condition based upon the signless Laplacian spectral radius to guarantee that a graph $G$ contains a spanning tree with leaf degree at most $k$. Finally, we construct some extremal graphs to claim all the bounds obtained in this paper are sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-11T10:28:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C05",
      "90B99"
    ],
    "keywords": [
      "signless laplacian spectral radius",
      "spanning tree",
      "diagonal degree matrix",
      "maximum leaf degree",
      "signless laplacian matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}