Graph Theory Questions Long
A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. In other words, it is a graph where each vertex is directly connected to every other vertex in the graph.
A complete graph is denoted by the symbol Kn, where n represents the number of vertices in the graph. For example, K3 represents a complete graph with 3 vertices, K4 represents a complete graph with 4 vertices, and so on.
In a complete graph, the number of edges can be calculated using the formula n(n-1)/2, where n is the number of vertices. This formula is derived from the fact that each vertex is connected to every other vertex, resulting in n-1 edges for each vertex. Since there are n vertices in total, the total number of edges is n(n-1)/2.
Complete graphs have several interesting properties. Firstly, they are symmetric, meaning that if there is an edge connecting vertex A to vertex B, there is also an edge connecting vertex B to vertex A. Secondly, complete graphs are simple graphs, meaning that there are no self-loops or multiple edges between the same pair of vertices.
Complete graphs are often used as a theoretical model in graph theory to study various properties and algorithms. They serve as a benchmark for comparison with other types of graphs and provide a foundation for understanding more complex graph structures.
In real-world applications, complete graphs are not commonly encountered due to their high connectivity. However, they can be useful in certain scenarios such as network design, where all nodes need to be directly connected to each other. Additionally, complete graphs are used in social network analysis to represent a fully connected network of individuals or entities.
In conclusion, a complete graph is a graph in which every pair of distinct vertices is connected by a unique edge. It is denoted by Kn, where n represents the number of vertices. Complete graphs have symmetric edges, are simple graphs, and serve as a theoretical model in graph theory. While not commonly encountered in real-world applications, they have their uses in certain scenarios.