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ConnectedGraphQ

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Test whether a graph is connected (strongly connected for directed graphs)

ConnectedGraphQ[g]
{ConnectedGraphQ[g], Total[VertexDegree[g]] ≥ (VertexCount[g] - 1) / 2} → {False, Total[VertexDegree[g]] >= (-1 + VertexCount[g])/2}

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