The maximal and the biggest graph

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What is the difference between the maximal graph and the biggest graph in graph theory?

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In the context you are probably asking for (although it is a guess) "maximal" often refers to maximal under inclusion and "biggest" to the one with most elements.

For example a maximal clique in a graph is a clique which is not contained in any bigger clique (cannot be extended to a bigger clique by adding one vertex). Whereas the largest/biggest clique will be a clique with largest number of vertices.

There can be many maximal cliques in a graph (and they can have different number of vertices). There can also be many largest cliques (and they will all have the same number of vertices). Every largest clique is maximal, but not vice versa.