Let κ′(G), μn−1(G) and μ1(G) denote the edge-connectivity, the algebraic connectivity and the Laplacian spectral radius of G, respectively. In this paper, we prove that for integers k≥2 and r≥2, and any simple graph G of order n with minimum degree δ≥k, girth g≥3 and clique number ω(G)≤r, the edge-connectivity κ′(G)≥k if μn−1(G)≥(k−1)nN(δ,g)(n−N(δ,g)) or if μn−1(G)≥(k−1)nφ(δ,r)(n−φ(δ,r)), where 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, and φ(δ,r)=max⁡{δ+1,⌊rδr−1⌋}. Analogue results involving μn−1(G) and μ1(G)μn−1(G) to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in Liu et al. (2013) 22, Liu et al. (2019) 20, Hong et al. (2019) 15, Liu et al. (2019) 21 and Abiad et al. (2018) 1 are improved or extended.