Edge-connectivity and (signless) Laplacian eigenvalue of graphs

In this paper, we first show that if the second smallest Laplacian eigenvalue of a graph is no less than (k−1)n(δ+1)(n−1−δ) or the second largest signless Laplacian eigenvalue of a graph is no more than 2δ−(k−1)n(δ+1)(n−1−δ), then the graph is k-edge-connected, where δ is the minimum degree of the graph and n is the order of the graph. Also, we give a Laplacian eigenvalue condition and a signless Laplacian eigenvalue condition for a graph to be k-edge-connected involving the girth g of the graph, respectively. Our results generalize some known results.