Let　τ(G),　λ2(G),　μn-1(G)　and　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≥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)≥k,　which　confirms　a　conjecture　of　Liu,　Hong　and　Lai,　and　also　implies　a　conjecture　of　Cioabǎ　and　Wong.　©　2014　Elsevier　Inc.