{
  "id": "2311.12417",
  "version": "v1",
  "published": "2023-11-21T08:09:59.000Z",
  "updated": "2023-11-21T08:09:59.000Z",
  "title": "Eigenvalues and spanning trees with constrained degree",
  "authors": [
    "Chang Liu",
    "Hao Li",
    "Jianping Li"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we study some spanning trees with bounded degree and leaf degree from eigenvalues. For any integer $k\\geq2$, a $k$-tree is a spanning tree in which every vertex has degree no more than $k$. Referring to the technique shown in [Eigenvalues and $[a,b]$-factors in regular graphs, J. Graph Theory. 100 (2022) 458-469], for a $r$-regular graph $G$, we provide an upper bound for the fourth largest eigenvalue of $G$ to guarantee the existence of a $k$-tree. Let $T$ be a spanning tree of a connected graph. The leaf degree of $T$ is the maximum number of leaves attached to $v$ in $T$ for any $v\\in V(T)$. For a $t$-connected graphs, we prove a tight sufficient condition for the existence of a spanning tree with leaf degree at most $k$ in terms of spectral radius. This generalizes a result of Theorem 1.5 in [Spectral radius and spanning trees of graphs, Discrete Math. 346 (2023) 113400]. Finally, for a general graph $G$, we present two sufficient conditions for the existence of a spanning tree with leaf degree at most $k$ via the Laplacian eigenvalues of $G$ and the spectral radius of the complement of $G$, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-21T08:09:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C70"
    ],
    "keywords": [
      "spanning tree",
      "leaf degree",
      "constrained degree",
      "spectral radius",
      "regular graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}