Let c(G), g(G), ω(G) and μn−1(G) denote the number of components, the girth, the clique number and the second smallest Laplacian eigenvalue of the graph G, respectively. The strength η(G) and the fractional arboricity γ(G) are defined byη(G)=minF⊆E(G)|F|c(G−F)−c(G)andγ(G)=maxH⊆G|E(H)||V(H)|−1, where the optima are taken over all edge subsets F and all subgraphs H whenever the denominator is non-zero, respectively. Nash-Williams and Tutte proved that G has k edge-disjoint spanning trees if and only if η(G)≥k; and Nash-Williams showed that G can be covered by at most k forests if and only if γ(G)≤k. In this paper, for integers r≥2, s and t, and any simple graph G of order n with minimum degree δ≥2st and either clique number ω(G)≤r or girth g≥3, we prove that if μn−1(G)>2s−1tφ(δ,r) or μn−1(G)>2s−1tN(δ,g), then η(G)≥st, where φ(δ,r)=max{δ+1,⌊rδr−1⌋} and N(δ,g) is the Moore bound on the smallest possible number of vertices such that there exists a δ-regular simple graph with girth g. As corollaries, sufficient conditions on μn−1(G) such that G has k edge-disjoint spanning trees are obtained. Analogous result involving μn−1(G) to characterize fractional arboricity of graphs with given clique number is also presented. Former results in Liu et al. (2014) [17] and Hong et al. (2016) [11] are extended, and the result in Liu et al. (2019) [18] is improved.