Below is a figure from this paper, which analyzes the algorithmic problem of calculating maximal cliques of a graph.
Unfortunately, no citation or elaboration is given for this graph, other than the claim that is has an exponential number of cliques.
My question is: is the below graph a standard example in Graph Theory that has been used before (if so, where), and does there exist a reference for the proof that it has an exponential number of cliques?
That particular graph of course does not have an exponential number of cliques: it has $16$ maximal cliques (all with $4$ vertices, so they are also all maximum cliques). To talk about an exponential number of cliques, we need to generalize this example to a family of graphs.
The generalization suggested by the notation is the cocktail party graph: this has $2n$ vertices $u_1, u_2, \dots, u_n, v_1, v_2, \dots, v_n$, and all vertices are adjacent except for the pairs $\{u_i, v_i\}$ for $i=1,\dots,n$. (The terminology is meant to evoke $n$ couples that go to a party; each person shakes hands with everyone there other than their partner.)
In the cocktail party graph, $u_i$ and $v_i$ cannot simultaneously be in a clique. However, every clique that contains neither $u_i$ nor $v_i$ can be extended by adding either one of those vertices. So there are $2^n$ maximal cliques, all with $n$ vertices, obtained by choosing one vertex from each pair $\{u_i, v_i\}$.
This is a standard example of a graph with exponentially many cliques, and I don't think it needs a reference. But if you wanted to cite something, the Moon-Moser paper On cliques in graphs proves that an $n$-vertex graph can have at most $3^{n/3}$ maximal cliques, and gives a related extremal example: take $n/3$ triples of vertices, and connect all vertices that are in different triples.