Clique number

Definition

The clique number of a graph is defined as the size of the largest complete subgraph contained within it. This concept is essential for understanding the structure of graphs, particularly in identifying groups of vertices that are all directly connected to each other. The clique number provides insights into the connectivity and clustering properties of graphs, which can relate to various types such as bipartite, complete, and regular graphs.

5 Must Know Facts For Your Next Test

1. The clique number is denoted by $$ ext{clique}(G)$$ for a graph $$G$$.
2. In a complete graph with $$n$$ vertices, the clique number is equal to $$n$$, as every vertex connects to every other vertex.
3. In bipartite graphs, the maximum clique number is at most 2 since no two vertices in the same set are adjacent.
4. For regular graphs, the clique number can be influenced by the degree of each vertex and how the vertices are arranged.
5. Determining the clique number can be computationally challenging and is often used in optimization problems related to social networks and clustering.

Review Questions

• How does the concept of clique number help in understanding the connectivity of graphs?
  • The clique number provides a clear indication of how interconnected the vertices within a graph are. By identifying the largest complete subgraph, we can assess how well vertices group together, which reflects overall connectivity. This understanding is crucial for analyzing network structures where strong interconnections indicate tighter relationships among elements.
• Compare the clique numbers of complete graphs and bipartite graphs and explain their implications.
  • Complete graphs have a clique number equal to their total number of vertices since every vertex is directly connected to all others. In contrast, bipartite graphs have a maximum clique number of 2, as they consist of two sets with no intra-set connections. This difference illustrates how structure influences connectivity; complete graphs signify high interconnectivity while bipartite graphs indicate more segregated relationships.
• Evaluate the significance of determining the clique number in practical applications such as social networks and clustering.
  • Determining the clique number has significant implications in fields like social networks and clustering because it reveals tightly-knit groups within larger populations. Identifying these groups allows researchers to understand community structures and interactions better. In social networks, knowing where cliques exist helps in analyzing influence patterns and information spread, thus enhancing our ability to design targeted strategies for engagement or intervention.

Related terms

Complete Graph: A graph in which every pair of distinct vertices is connected by a unique edge, making its clique number equal to the total number of vertices.

Bipartite Graph: A graph whose vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent; its clique number is at most 2.

Regular Graph: A graph where each vertex has the same degree; its clique number can vary based on the degree and overall structure of the graph.