The Complete Graph | Properties, Examples, and Applications in Graph Theory and Network Analysis

Kn (complete graph on n vertices)

In graph theory, the complete graph on n vertices, often denoted as Kn, is a simple graph that contains all possible edges between its n vertices

In graph theory, the complete graph on n vertices, often denoted as Kn, is a simple graph that contains all possible edges between its n vertices. In other words, it is a graph where every pair of distinct vertices is connected by an edge.

Properties of a complete graph:

1. Vertices: A complete graph on n vertices has n nodes or vertices.
2. Edges: It has a total of n(n-1)/2 edges since each vertex is connected to all other n-1 vertices.
3. Degree: In a complete graph, every vertex has a degree of n-1 since it is connected to all the other n-1 vertices.
4. Connectedness: A complete graph is always a connected graph because every pair of vertices is connected by an edge.
5. Symmetry: The edges in a complete graph are symmetric. If vertex A is connected to vertex B, then vertex B is also connected to vertex A.
6. Diameter: The diameter of a complete graph is 1, meaning that the distance between any two vertices is at most 1.

Examples:
1. K1: A complete graph with only one vertex, which has no edges.
2. K2: A complete graph with two vertices, connected by a single edge.
3. K3: A complete graph with three vertices, where each vertex is connected to the other two vertices by edges forming a triangle.
4. K4: A complete graph with four vertices, where each vertex is connected to the other three vertices by edges forming a square.

Applications:
Complete graphs are primarily used in studying graph theory and network analysis. They can be used as models for social networks, communication networks, transportation networks, and any situation where all pairs of entities are connected or have potential connections. They also provide a basis for understanding graph algorithms and concepts like connectivity, clustering, and shortest paths.

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