EdgeConnectivity returns the edge connectivity of a graph, that is the minimum number of edges whose removal disconnects the graph. A set of such edges is called an edge-cut.  You can use the IsCutSet command to test whether a set of edges is an edge-cut.

VertexConnectivity returns the vertex connectivity of a graph, that is the minimum number of vertices whose removal disconnects the graph.

By an elementary theorem of graph theory, the vertex connectivity of a graph is less than or equal to the edge connectivity, which is less than or equal to the minimum degree.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{Graph}\left(\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{1\,4\right\}\,\left\{3\,4\right\}\,\left\{4\,5\right\}\,\left\{5\,6\right\}\,\left\{4\,6\right\}\right\}\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 8\; edge(s)}$

$\mathrm{DrawGraph}\left(G\right)$

$\mathrm{EdgeConnectivity}\left(G\right)$

$2$

$\mathrm{VertexConnectivity}\left(G\right)$

$1$

$\mathrm{MinimumDegree}\left(G\right)$

$2$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$

$P≔\mathrm{PetersenGraph}\left(\right)$

$P≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}$

$\mathrm{VertexConnectivity}\left(P\right)$

$3$

$\mathrm{EdgeConnectivity}\left(P\right)$

$3$

$\mathrm{MinimumDegree}\left(P\right)$

$3$