graph theory - clique graph

Question

I am trying to understand the concept of clique graph. So I found this page. But I do not understand the example and what "graph intersection" is.

Can somebody explain to me why $K_4$ is a clique graph of $G$?

All I understand so far is that this particular graph $G$ has 4 cliques they are all $K_3$.

Answer 1

If we have some collection of sets, the intersection graph of the sets is given by representing each set by a vertex and then adding edges between any sets that share an element. (Wikipedia has a nice picture in the intersection graph article.)

The clique graph is the intersection graph of the maximal cliques.

As you note, $G$ has four maximal cliques.

Each of the four cliques shares at least one vertex with each of the other cliques, so in this case the intersection graph is complete.

Answer 2

Suppose you go through and circle all of the maximal cliques of $G$ (which, as you noted, are all $K_3$s). These will be the vertices of the clique graph. When two (or more) circles overlap at a vertex in the original graph, those vertices will have an edge between them in the clique graph.