Graph Theory - Complete Graphs

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Complete Graphs

A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. In other words, in a complete graph, every vertex is adjacent to every other vertex.

Complete graphs are denoted by the symbol K_n, where n represents the number of vertices in the graph. A complete graph with n vertices contains n(n-1)/2 edges, making it one of the densest possible graphs.

Example

Consider a complete graph with 5 vertices K_5. The graph would have the following edges −

E = {(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

Here, every vertex is connected to every other vertex in the graph.

Properties of Complete Graphs

Complete graphs have several unique properties that distinguish them from other types of graphs. These properties plays an imporrant role in graph theory and its applications.

High Degree of Vertices

In a complete graph with n vertices, each vertex has a degree of n-1. This means that each vertex is connected to every other vertex in the graph.

Example: In a complete graph K_5 (a graph with 5 vertices), each vertex will have a degree of 4 since each vertex is connected to the other 4 vertices.

Number of Edges

A complete graph with n vertices has exactly n(n-1)/2 edges. This can be derived from the fact that each vertex connects to every other vertex once, and since the edges are undirected, the total number of edges is half the product of n and n-1.

Example: A complete graph K_6 (6 vertices) will have 6(6-1)/2 = 15 edges.

Complete Graphs Are Simple Graphs

A complete graph is a simple graph, meaning that it does not contain any loops (edges that connect a vertex to itself) or multiple edges (edges that connect the same pair of vertices). In K_n, each pair of distinct vertices is connected by exactly one edge.

Complete Graphs Are Connected

In a complete graph, all vertices are reachable from any other vertex. Therefore, a complete graph is always connected, meaning there is a path between any two vertices in the graph.

Complete Graphs Are Hamiltonian

A Hamiltonian cycle is a cycle that visits every vertex exactly once and returns to the starting vertex. Complete graphs with more than two vertices are always Hamiltonian, meaning that such graphs always contain a Hamiltonian cycle.

Complete Graphs Are Eulerian

An Eulerian cycle is a cycle that uses every edge of the graph exactly once and returns to the starting vertex. A complete graph with an even number of vertices is Eulerian because every vertex has an even degree, which is a necessary condition for the existence of an Eulerian cycle.

Graph Diameter

The diameter of a graph is the longest shortest path between any two vertices. In a complete graph, the diameter is always 1 because every pair of vertices is directly connected by an edge, making the shortest path between any two vertices a single edge.

Graph Radius

The radius of a graph is the minimum eccentricity of any vertex, where eccentricity is the greatest distance from a vertex to any other vertex. In a complete graph, the radius is 1, as every vertex is adjacent to every other vertex, so the eccentricity of any vertex is 1.

Applications of Complete Graphs

Complete graphs have several real-world applications in various fields such as computer science, network theory, and operations research.

Network Design

In network design, complete graphs can be used to model fully connected networks where every node (e.g., a computer or a server) is directly connected to every other node. This is particularly relevant in scenarios where high fault tolerance and redundancy are required.

Optimization Problems

Complete graphs are frequently used in optimization problems, such as the Traveling Salesman Problem (TSP), where the goal is to find the shortest possible route that visits each vertex (city) exactly once and returns to the starting point. Complete graphs are used in these problems because they represent all possible routes between vertices.

Parallel Computing

In parallel computing, complete graphs are often used to represent communication patterns between processors. Each processor is connected to every other processor, allowing for efficient data exchange in highly parallel systems.

Social Networks

In social network analysis, a complete graph can represent a social network where every individual is connected to every other individual. This type of graph is useful for modeling highly connected social networks with no missing connections.

Graph Coloring

Complete graphs also play a role in graph coloring problems, where the task is to assign colors to vertices such that no two adjacent vertices share the same color. In a complete graph K_n, it requires n different colors, as every vertex is adjacent to every other vertex.

Complete Graphs in Special Cases

Complete graphs can also be analyzed in special cases based on their size or certain graph properties. These include specific properties for even and odd numbers of vertices.

Complete Graph with Even Number of Vertices

If a complete graph has an even number of vertices n, then each vertex has an odd degree, and the graph is Eulerian. This makes it possible to find an Eulerian cycle in a K_n where n is even.

Complete Graph with Odd Number of Vertices

If a complete graph has an odd number of vertices, each vertex has an even degree, but the graph is not Eulerian. However, it is still Hamiltonian and connected.

Complete Graph with 2 Vertices

A complete graph with 2 vertices, K_2, is a simple graph with one edge connecting the two vertices. This is the smallest possible complete graph.

Complete Graph with 3 Vertices

A complete graph with 3 vertices, K_3, forms a triangle. It is both Hamiltonian and Eulerian, as it contains both a Hamiltonian cycle and an Eulerian cycle.

Complete Graphs Computational Complexity

The study of the computational complexity of problems involving complete graphs is essential in understanding how graph algorithms behave on complete graphs.

Time Complexity of Algorithms on Complete Graphs

For algorithms like depth-first search (DFS), breadth-first search (BFS), and Dijkstra's algorithm, the time complexity depends on the number of vertices and edges. In a complete graph, there are n(n-1)/2 edges, so these algorithms may have different time complexities compared to sparse graphs.

Computational Complexity of Graph Problems

Many graph problems, such as the Traveling Salesman Problem (TSP) and Maximum Matching, are computationally hard on complete graphs. For TSP, the problem is NP-hard, meaning that finding an optimal solution for large complete graphs can take exponential time.