Note on edge-disjoint spanning trees and eigenvalues

Let τ(G)τ(G), λ2(G)λ2(G), μn−1(G)μn−1(G) and ρ2(G)ρ2(G) be the maximum number of edge-disjoint spanning trees, the second largest adjacency eigenvalue, the algebraic connectivity, and the second largest signless Laplace eigenvalue of G, respectively. In this note, we prove that for any graph G   with minimum degree δ≥2kδ≥2k, if λ2(G)<δ−2k−1δ+1 or μn−1(G)>2k−1δ+1 or ρ2(G)<2δ−2k−1δ+1, then τ(G)≥kτ(G)≥k, which confirms a conjecture of Liu, Hong and Lai, and also implies a conjecture of Cioabă and Wong.