# (A puzzle) The largest size cliques in n-grouping graphs

• ygolo
In summary, for any given n-grouping G, there are (n!)^(n-1) complementary n-groupings, and the total number of such cliques is (n^2)!/(n!)^(n+1). For n=2, there is only 1 clique--the whole graph of three vertices.
ygolo
(I apologize in advance for the abuse of the word "group" in this problem. It does not mean a group in the mathematical sense).

Imagine you have n^2 objects that need to be grouped in n groups of size n each. Let's call one such arrangement an "n-grouping."

You can create n-groupings in (n^2)!/(n!)^(n+1) ways (check me on this).

Now suppose you have one of these n-groupings, G1, and you would like to find a "complementary" n-grouping, G2. What I mean by this is that every group in G2 has exactly 1 object from each group of G1 (and vice versa--I believe it provable that if G2 is complementary to G1, then G1 is complementary to G2--check me on this).

Now, for any given any n-grouping G, there are (n!)^(n-1) complementary n-groupings (check me on this too).

Now imagine a graph where all the n-groupings are vertices, and where there is an edge between two n-groupings if and only if the two n-groupings are complementary.

My questions are:
1) What is the size of the largest clique?
2) How many such cliques are there?

For n=2, there is only 1 clique--the whole graph of three vertices. (This seems obvious to me, but check me on this one too)

Thanks in advance!1) The size of the largest clique is (n!)^(n-1).2) The number of such cliques is (n^2)!/(n!)^(n+1).

## 1. What is a "clique" in n-grouping graphs?

A clique in n-grouping graphs refers to a subset of vertices that are all connected to each other. In other words, every vertex in the clique is connected to every other vertex in the clique.

## 2. How do you determine the largest size cliques in n-grouping graphs?

The largest size cliques in n-grouping graphs can be determined by using algorithms such as the Bron-Kerbosch algorithm or the maximal clique enumeration algorithm. These algorithms systematically search through all possible subsets of vertices to find the largest cliques.

## 3. What is the significance of studying the largest size cliques in n-grouping graphs?

Studying the largest size cliques in n-grouping graphs can provide insights into the structure and complexity of these graphs. It can also have practical applications in fields such as social network analysis, where cliques represent tightly-knit groups of individuals.

## 4. Can the largest size cliques in n-grouping graphs change?

Yes, the largest size cliques in n-grouping graphs can change depending on the graph's structure and the grouping of vertices. Different grouping methods can result in different largest cliques.

## 5. How can the largest size cliques in n-grouping graphs be used in real-world problems?

The largest size cliques in n-grouping graphs can be used in various real-world problems such as community detection, pattern recognition, and data clustering. In these applications, the cliques can help identify groups or patterns within a larger dataset.

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