Let　tau(G),　lambda(2)(G),　mu(n-1)(G)　and　rho(2)(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　delta　＞=　2k,　if　lambda(2)(G)　＜　delta　-　2k-1/delta+1　Or　mu(n-1)(G)　＞　2k-1/delta+1　or　rho(2)(G)　＜　2　delta　-　2k-1/delta+1,　then　tau(G)　＞=　k,　which　confirms　a　conjecture　of　Liu,　Hong　and　Lai,　and　also　implies　a　conjecture　of　Cioabg　and　Wong.　(C)　2014　Elsevier　Inc.　All　rights　reserved.