## Clique

### In graph theory, a clique is a subset of vertices in an undirected graph in which every pair of vertices is connected by an edge

In graph theory, a clique is a subset of vertices in an undirected graph in which every pair of vertices is connected by an edge. In other words, a clique is a complete subgraph of a graph. The size of a clique is the number of vertices it contains.

We can think of a clique as a group of people where every person is directly friends with every other person in the group. If there are n people in a clique, the maximum number of friendships in the group is given by the formula n(n-1)/2.

For example, consider a graph with 5 vertices: A, B, C, D, and E. If there is an edge between every pair of vertices, we have a clique of size 5. This means that every vertex is directly connected to the other four vertices, forming a complete subgraph.

Clique finding is an important problem in graph theory and has various applications in computer science, social networks, and data analysis. Identifying cliques helps in understanding relationships or communities within a graph and can be used for tasks such as finding groups of similar individuals or identifying densely connected areas in a network.

There are algorithms, such as the Bron-Kerbosch algorithm, that can be used to find cliques in a graph. These algorithms explore different subsets of vertices to determine if they form a clique. The problem of finding the maximum clique, which is the largest clique in a graph, is known to be NP-hard and computationally challenging for large graphs.

Overall, cliques are an important concept in graph theory that helps in understanding and analyzing relationships within graphs.

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