Maximal and Maximum Cliques

Question

I am working on an exercise based on this image. I have found the maximum clique size to be 4. I have a few questions on the concept of graph theory.

By definition, a clique is a complete subgraph where each pair of vertices are connected. Would this mean that if I was counting 3-cliques, (3,4,5), (3,4,6), (3,5,6), and (4,5,6) would count as 3-cliques? Or should I omit those subgraphs since they are part of the 4-clique.

Does every graph have only one maximum clique? Imagining it visually in my mind, I feel like it is possible to have more than one maximum clique.

One of the questions in the exercise asks if every graph with one or more nodes must have at least one clique. Is there such thing as a 2-clique (just an edge) or should every clique form a closed shape?

I can't seem to draw an instance of a 4-clique that does not have a 3-clique, so it is safe to assume that every 4-clique has at least one 3-clique? How would I go about checking for something like this on a larger scale?

Answer 1

First of all, all those 3-cliques you mentioned are indeed cliques.
As you said, a clique is a subgraph where all the nodes are connected to all the other nodes. So in your example, (3,4,5) is a clique, and so is (3,4,5,6), and so are (3) and (3,4) (which also answers some of your other questions).

Regarding maximum cliques, of course you could have more than 1 - for example, if you take the (3,4,5,6) from your graph, duplicate it to (3',4',5',6'), and connect 3 with 3', then you have 2 4-cliques in your graph, but no 5-cliques.

Also, any subgraph of a clique is also a clique, since every subgraph still satisfies the demand for all nodes being connected to all the other ones. For example in your graph, (3,4,5,6) is a 4-clique. Take any 3 nodes from there, and you shall get a 3-clique. Take 2, and you get a 2-clique. So in fact, not only every 4-clique has at least 1 3-clique in it, but it has exactly 4 3-cliques in it (that's 4C3).

Note, however, that sometimes cliques are defined as having 2 or more (or sometimes 3 or more) nodes, because anything smaller is a bit trivial, for lack of a better word.