Complement of a Graph in Discrete Mathematics: Understanding Graph Complementation

Learn about the complement of a graph in discrete mathematics. This guide explains how to construct the complement graph, its relationship to the original graph, and provides examples demonstrating calculations involving the number of vertices and edges in a graph and its complement.

Complement of a Graph in Discrete Mathematics

What is the Complement of a Graph?

The complement of a simple graph G (denoted G') is another graph with the same set of vertices as G. However, two vertices are connected by an edge in G' if and only if they are not connected by an edge in G. Think of it as highlighting the missing connections in the original graph.

Relationship Between a Graph and its Complement

  • Vertices: The number of vertices in G and G' are the same: |V(G)| = |V(G')|.
  • Edges: The total number of edges in G and G' together equals the number of edges in a complete graph (a graph where every pair of vertices is connected) with the same number of vertices. Formally: |E(G)| + |E(G')| = n(n - 1) / 2, where n is the number of vertices.

Key Points about Graph Complements

  • The size (number of vertices) of G and G' are the same.
  • The order (number of vertices) of G and G' are the same.

Examples: Finding the Number of Edges in a Complement Graph

Example 1: Calculating Edges in the Complement

(This example, calculating the number of edges in the complement graph given the number of vertices and edges in the original graph, is provided in the original text and should be included here. The solution should clearly show the application of the formula relating the number of edges in a graph and its complement.)

Example 2: Calculating Vertices from Edges

(This example, calculating the number of vertices in the original graph given the number of edges in the original graph and its complement, is provided in the original text and should be included here. The solution should clearly show the application of the formula and the use of the quadratic formula to solve for the number of vertices.)

Example 3: Finding the Order of a Graph

(This example, calculating the number of vertices in graph G given the number of edges in G and its complement G', is provided in the original text and should be included here. The solution should clearly show the application of the formula and the use of the quadratic formula.)

Conclusion

The complement of a graph provides a way to study the relationships between connected and disconnected vertices within a graph. The relationship between the number of edges in a graph and its complement is a useful tool in graph theory.