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：对于包含$n$个顶点的简单图，其可能包含大量最大团。但问题在于：它最多能包含多少个最大团？我们将定义素图和复合图，并证明当$n \ge 15$时，具有最多最大团数量的图必须是复合图。此外，我们将给出一个边数界限，并由此证明：若复合图的某个因子满足$\omega(G_i) \ge 5$，则该图不可能拥有最多最大团数量。基于此结论，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最大团数量，其中$c\in\{1,\frac{4}{3},2\}$，其取值取决于$n \text{ mod } 3$。