Calculating the maximum number of maximum cliques for simple graphs

TL;DR

This paper investigates the maximum number of maximum cliques in simple graphs, establishing that for sufficiently large graphs, the extremal graphs are composite and characterizing their structure based on vertex partitioning.

Contribution

It introduces the concepts of prime and composite graphs and determines the structure of graphs with the maximum number of maximum cliques for large n.

Findings

Graphs with maximum maximum cliques are composite for n ≥ 15.

Maximum number of maximum cliques is achieved by graphs with a specific partition structure.

The maximum count follows a formula involving powers of 3 and a constant c depending on n mod 3.

Abstract

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 $n \text{ mod } 3$.