Edge-disjoint spanning trees and eigenvalues of graphs

It is proven that a graph of order n   with minimum degree δ⩾2kδ⩾2k has k   edge-disjoint spanning trees if its second largest eigenvalue is less than δ−2k−1δ+1−O(1/n). This extends earlier results of Cioabă and Wong and improves results of Gu, Lai, Li and Yao, and also confirms approximately their conjectures for sufficiently large graphs with respect to the minor term. We also conjecture that the minor term can be removed.