{
  "annotation_id": "8ae0a22e-fad6-4c14-a780-9b656f75c285",
  "date_created": "2026-02-17T05:53:12.331000Z",
  "date_modified": "2026-02-17T05:53:12.331000Z",
  "file_hash": "699084416fa2c00162006ce6b92e876a63477cf7c6141383b23c26973d5e8de9",
  "private": false,
  "record": {
    "abstract": "Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)| \u003e \\frac{d-1}{d}(|V(G)| - 1)$, such that if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Let $D(G)$ be the degree diagonal matrix of $G$. We denote $\\lambda_i$ and $\\mu_i$ as the $i$th largest eigenvalue of the adjacency matrix $A(G)$ of $G$ and the Laplacian matrix $L(G) = D(G) - A(G)$ of $G$ for $i = 1, 2, \\ldots, n$, respectively. In this paper, we investigate the relationship between Laplacian eigenvalues and property $P(k, \\delta)$. Let $t$ be a positive integer, and define $\\mathcal{G}_t$ as the set of simple graphs such that each $G \\in \\mathcal{G}_t$ contains at least $t+1$ non-empty disjoint proper subsets $V_1, V_2, \\ldots, V_{t+1}$ satisfying $V(G) \\setminus \\bigcup_{i=1}^{t+1} V_i \\neq \\emptyset$ and edge connectivity $\\kappa\u0027(G) = e(V_i, V(G) \\setminus V_i)$ for any $i = 1, 2, \\ldots, t+1$. For the class of graphs $\\mathcal{G}_1$ with minimum degree $\\delta$, we provide a sufficient condition involving the third smallest Laplacian eigenvalue $\\mu_{n-2}(G)$ for a graph $G\\in \\mathcal{G}_1$ to have property $P(k, \\delta)$. Similarly, for the class of graphs $\\mathcal{G}_2$ with minimum degree $\\delta$, we establish a corresponding sufficient condition involving the fourth smallest Laplacian eigenvalue $\\mu_{n-3}(G)$ for a graph $G\\in \\mathcal{G}_2$ to have property $P(k, \\delta)$. Furthermore, we extend the spectral conditions for all the results about $\\mu_{n-2}(G)$, $\\mu_{n-3}(G)$ and $\\lambda_2(G)$ to the general graph matrices $aD(G) + A(G)$ and $aD(G) + bA(G)$.",
    "arxiv_id": "2601.07893",
    "authors": [
      "Yongbin Gao",
      "Ligong Wang"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Laplacian eigenvalue conditions for edge-disjoint spanning trees and a forest with constraints",
    "url": "https://arxiv.org/abs/2601.07893",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "d14333c4-eca5-427f-8add-f5e5e443bec6",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}