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Is there a difference between a complete graph and a clique topology? As far as I understand, both refer to graphs in which every possible edge between any two vertices is present. Is there a subtle difference between these two concepts?

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A complete graph is a graph with every possible edge; a clique is a graph or subgraph with every possible edge. That is, one might say that a graph "contains a clique" but it's much less common to say that it "contains a complete graph".

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