Let be a connected graph with minimum degree and vertex-connectivity . The graph is -connected if , maximally connected if , and super-connected if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we present sufficient conditions for a graph with given minimum degree to be -connected, maximally connected, or super-connected in terms of the number of edges, the spectral radius of the graph, and its complement, respectively. Analogous results for triangle-free graphs with given minimum degree to be -connected, maximally connected, or super-connected are also presented.

中文翻译：

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图的连通，最大连通和超连通的充分条件

设一个具有最小度和顶点连通性的连通图。该图已连接-如果，如果连接最大，如果每个最小顶点切割都隔离了最小度顶点，则超级连接。在本文中，我们提出了与给定的最小程度的曲线图的充分条件被-连接的，最大程度地连接，或者超连接的分别的边缘的数目，该曲线图的谱半径，和它的补码，计。还给出了具有给定的最小度数的无三角形图的相似结果-连接，最大连接或超级连接。