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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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A clique is an induced subgraph that is itself complete graph.
What's an induced subgraph?
Ans: Given a graph, we pick a set of vertices and construct a subgraph. But now if we also select all edges that were incident on the selected set of vertices in the original graph, then we construct an induced subgraph. If this induced subgraph has an edge between all vertex pairs then its is complete.
So I think every complete graph can be thought of as an induced subgraph of some larger graph and hence every complete graph can be considered as a clique.

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