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For a simple graph of 1 node/vertex with no edges (null graph), what would the complement of this graph be? Would it be a loop with itself or a null graph with one vertex? Or neither?

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I think we keep the "simple" property of a graph under complementation. Hence, the complement of the simple graph, $G$, consisting of one vertex would be $G$ itself. –  pedrosuavo Jan 13 '13 at 0:37

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up vote 1 down vote accepted

It would be the same graph, since a simple graph cant have loops, and a complement graph has the same vertices as the original.

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