Clique number

Definition

The clique number of a graph is the size of the largest complete subgraph, or clique, that can be found within it. It represents how many vertices can be connected in a fully interconnected way, meaning every vertex in that group is adjacent to every other vertex. Understanding the clique number helps analyze the structure and properties of graphs, especially in contexts involving dense connections or relationships.

5 Must Know Facts For Your Next Test

1. The clique number is denoted as $$ ext{cl}(G)$$ for a graph $$G$$ and is critical in studying graph properties and characteristics.
2. Mantel's Theorem states that any triangle-free graph on $$n$$ vertices has at most $$\frac{n^2}{4}$$ edges, leading to implications about its clique number being at most 2.
3. In dense graphs, the clique number can provide insights into potential subsets of highly connected nodes, relevant in social networks and biological interactions.
4. Graphs with higher clique numbers often indicate a greater level of interconnectivity, which can affect coloring problems and the chromatic number.
5. Finding the clique number is an NP-hard problem in general, meaning there is no known efficient algorithm to determine it for all graphs.

Review Questions

• How does Mantel's Theorem relate to the concept of clique number in triangle-free graphs?
  • Mantel's Theorem highlights a crucial relationship between triangle-free graphs and their maximum edge count, indicating that such graphs have a clique number of at most 2. This shows that while they can be densely connected through edges, they cannot contain any triangles. As a result, this theorem helps establish upper bounds on the clique number based on the graph's properties.
• Discuss how understanding the clique number can impact graph coloring problems.
  • Understanding the clique number directly affects graph coloring since larger cliques require more colors for proper vertex coloring. If a graph has a large clique number, it means there are many interconnected vertices that must all be assigned different colors. Consequently, knowing the clique number allows for better strategies in determining the chromatic number, which is critical for scheduling problems and resource allocation in various applications.
• Evaluate the challenges associated with computing the clique number and its implications for practical applications in real-world networks.
  • Computing the clique number poses significant challenges due to its classification as an NP-hard problem. This complexity affects practical applications like social network analysis and biological research, where determining tightly-knit groups or communities is essential. The inability to quickly compute the clique number can lead to difficulties in modeling real-world systems efficiently, potentially impacting insights drawn from network data and hindering optimal decision-making.

Related terms

Graph: A collection of vertices (or nodes) and edges (connections between pairs of vertices), often used to model relationships in various fields.

Complete Graph:

A type of graph where there is an edge connecting every pair of vertices, which means the clique number equals the total number of vertices.

Independence Number:

The size of the largest set of vertices in a graph, none of which are adjacent to each other; it provides a contrast to the concept of clique number.