A complete graph is an undirected graph in which every pair of distinct vertices is connected by a unique edge. In other words, every vertex in a complete graph is adjacent to all other vertices. A complete graph is denoted by the symbol K_n, where n is the number of vertices in the graph.
Characteristics of Complete Graph:
The main characteristics of a complete graph are:
- Connectedness: A complete graph is a connected graph, which means that there exists a path between any two vertices in the graph.
- Count of edges: Every vertex in a complete graph has a degree (n-1), where n is the number of vertices in the graph. So total edges are n*(n-1)/2.
- Symmetry: Every edge in a complete graph is symmetric with each other, meaning that it is un-directed and connects two vertices in the same way.
- Transitivity: A complete graph is a transitive graph, which means that if vertex A is connected to vertex B and vertex B is connected to vertex C, then vertex A is also connected to vertex C.
- Regularity: A complete graph is a regular graph, meaning that every vertex has the same degree.
How to Identify Complete Graph?
To identify a complete graph, you need to check if every vertex in the graph is connected to every other vertex. Here are two methods for identifying a complete graph:
- Check the degree of each vertex: In a complete graph with n vertices, every vertex has degree n-1. So, if you can determine that every vertex in the graph has degree n-1, then the graph is a complete graph.
- Check the number of edges: A complete graph with n vertices has n*(n-1)/2 edges. So, if you can count the number of edges in the graph and verify that it has n*(n-1)/2 edges, then the graph is a complete graph.
Note: These methods are effective if it s ensured that the graph does not have any cycle.
Applications of Complete Graph :
Complete graphs have many applications in various fields, including:
- Transportation networks: Complete graphs can be used to represent transportation networks, such as motorways or aircraft routes, where every point is connected to every other location directly.
- Network analysis: The characteristics of other graphs, such as clustering, connectedness, or shortest path length, can be measured against the attributes of complete graphs.
- Optimization: Complete graphs are used in optimization problems, such as maximum clique, which involves finding the widest subset of vertices in a graph that are all mutually adjacent.