Prove that the sum of the degree of all the vertices in a graph G is equal to twice the number of edges in G.

Introduction

In the context of discrete structures, a graph is a mathematical structure used to model pairwise relations between objects. A graph is made up of vertices (or nodes) and edges (or arcs). Each edge connects two vertices, indicating a relationship between them.

The degree of a vertex in a graph is the number of edges that are incident to it. In other words, it is the number of edges that connect to the vertex. For example, in a graph with three vertices A, B, and C, where A is connected to B and B is connected to C, the degree of vertex B is 2, while the degrees of vertices A and C are 1.

Theorem Explanation

The theorem states that the sum of the degrees of all the vertices in a graph G is equal to twice the number of edges in G. This is because each edge in the graph contributes exactly 2 to the total degree count, as it is connected to two vertices. Therefore, when we sum up the degrees of all vertices, we are essentially counting each edge twice, hence the sum is twice the number of edges.

Proof of Theorem

Let's consider a graph G with n vertices and m edges. Each edge in the graph is connected to two vertices, therefore, it contributes 2 to the total degree count. If we sum up the degrees of all vertices, we are essentially counting each edge twice. Hence, the sum of the degrees of all vertices is 2m, which is twice the number of edges.

Example

Let's consider a simple graph with 4 vertices (A, B, C, D) and 4 edges (AB, BC, CD, DA). The degree of each vertex is 2 (since each vertex is connected to two other vertices). Therefore, the sum of the degrees of all vertices is 2 + 2 + 2 + 2 = 8. On the other hand, the number of edges in the graph is 4, and twice this number is 8. Hence, the sum of the degrees of all vertices is indeed equal to twice the number of edges.

Conclusion

In conclusion, the theorem that the sum of the degrees of all the vertices in a graph G is equal to twice the number of edges in G holds true. This is because each edge in the graph contributes exactly 2 to the total degree count, as it is connected to two vertices. Therefore, when we sum up the degrees of all vertices, we are essentially counting each edge twice, hence the sum is twice the number of edges.

Programming (Optional)

Here is a simple Python program that takes a graph as input and verifies the theorem by calculating the sum of the degrees of all vertices and comparing it to twice the number of edges.

def verify_theorem(graph):
    # Calculate the sum of the degrees of all vertices
    degree_sum = sum(len(edges) for edges in graph.values())

    # Calculate twice the number of edges
    edge_count = sum(len(edges) for edges in graph.values()) // 2
    twice_edge_count = 2 * edge_count

    # Verify the theorem
    return degree_sum == twice_edge_count

# Test the function with a simple graph
graph = {
    'A': ['B', 'D'],
    'B': ['A', 'C'],
    'C': ['B', 'D'],
    'D': ['A', 'C'],
}
print(verify_theorem(graph))  # Output: True

In this program, the verify_theorem function calculates the sum of the degrees of all vertices by summing up the lengths of the edge lists for each vertex in the graph. It then calculates twice the number of edges by summing up the lengths of the edge lists for each vertex and dividing by 2 (since each edge is counted twice), and then multiplying by 2. The function returns True if the sum of the degrees of all vertices is equal to twice the number of edges, and False otherwise.

The test graph is a simple graph with 4 vertices and 4 edges, similar to the one used in the example above. The output of the function is True, verifying that the theorem holds for this graph.

Diagram

In this case, a diagram is not necessary as the concept and proof can be clearly understood without it. However, a diagram can be used to visually illustrate the concept and the example, if desired.

Summary

The sum of the degree of all the vertices in a graph G is equal to twice the number of edges in G.

Analogy

Imagine a group of friends where each person shakes hands with every other person. The number of handshakes represents the number of edges in the graph, and the degree of each person represents the number of handshakes they have. The sum of the degrees of all the people in the group will be equal to twice the number of handshakes.