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Let λ2(G) and τ (G) denote the second largest eigenvalue and themaximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of τ (G), Cioaba andWong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if λ2(G) < d - 2k-1/d+1, then τ (G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. Thus the conjecture remains open for k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree δ ≥ 2k ≥ 4, if λ2(G) < δ - 2k-1 δ+1, then τ (G) ≥ k. In this article, we prove that for a graph G with minimum degree δ, each of the following holds. (i) For k {2, 3}, if δ ≥ 2k and λ2(G) < δ - 2k-1/δ+1, then τ (G) ≥ k. (ii) For k ≥ 4, if δ ≥ 2k and λ2(G) < δ - 3k-1/δ+1, then τ (G) ≥ k. Our results sharpen theorems of Cioaba and Wong and give a partial solution to Cioaba and Wong's conjecture and Seymour's problem. We also prove that for a graph G with minimum degree δ ≥ k ≥ 2, if λ2(G) < δ - 2(k-1) δ+1, then the edge connectivity is at least k, which generalizes a former result of Cioab? a. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on τ (G) and edge connectivity. [ABSTRACT FROM AUTHOR]