A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will define prime and composite graphs, and we will show that if $n \ge 15$, then the grpahs with the maximum number of maximum cliques have to be composite. Moreover, we will show an edge bound from which we will prove that if any factor of a composite graph has $\omega(G_i) \ge 5$, then it cannot have the maximum number of maximum cliques. Using this we will show that the graph that contains $3^{\lfloor n/3 \rfloor}c$ maximum cliques has the most number of maximum cliques on $n$ vertices, where $c\in\{1,\frac{4}{3},2\}$, depending on $n \text{ mod } 3$.

翻译：在$n$个顶点上的简单图可能包含大量最大团，但其潜在的最大数量是多少？我们将定义素图与复合图，并证明当$n \ge 15$时，具有最多最大团数量的图必定是复合图。此外，我们将给出一个边界的约束，由此证明若复合图的任意因子满足$\omega(G_i) \ge 5$，则其不可能达到最大团的最大数量。基于此，我们将证明在$n$个顶点上，包含$3^{\lfloor n/3 \rfloor}c$个最大团的图具有最多的最大团数量，其中$c\in\{1,\frac{4}{3},2\}$，其取值取决于$n \text{ mod } 3$。