Problem 44 - Discrete Mathematics with Applications

Question

How many edges does an \(r\) -regular graph with \(n\) vertices have? (Hint: Use Exercise 35.)

Step-by-Step Solution

Answer

An $r$-regular graph with $n$ vertices has \(\frac{n \times r}{2}\) edges.

Step 1: Understand the relationship between degrees and edges

In a graph, the sum of the degrees of all vertices is equal to twice the number of edges (from Exercise 35). This is because each edge connects two vertices and contributes one to the degree of both vertices.

Step 2: Calculate the sum of degrees of all vertices in the r-regular graph with n vertices

Since we have an r-regular graph with n vertices, each vertex has a degree of r. This means that the sum of degrees of all vertices in the graph = n * r

Step 3: Use the relationship between degrees and edges to determine the number of edges in the r-regular graph

According to Exercise 35, the sum of degrees of all vertices in a graph = 2 * (number of edges). So, we have n * r = 2 * (number of edges).

Step 4: Solve for the number of edges

To find the number of edges, we can rearrange the equation from Step 3: number of edges = (n * r) / 2 Therefore, an r-regular graph with n vertices has (n * r) / 2 edges.

Key Concepts

Regular Graph

In graph theory, a regular graph is a type of graph where each vertex has the same number of connections, or edges, with other vertices. This uniform connection means every vertex holds the same degree, which is why regular graphs are sometimes referred to as "equally connected" graphs. Regular graphs can be further categorized based on their degree:

• r-regular graph: If each vertex in the graph has exactly \( r \) connections, the graph is called an r-regular graph.
• Examples: If each vertex has 2 connections, it's a 2-regular graph. Similarly, a 3-regular graph would mean each vertex connects to 3 others.

These graphs are used in various mathematical and real-world applications, such as network design and adjacency matrices. Understanding regular graphs helps provide a structured way to look at certain symmetrical properties within graphs.

Degree of a Vertex

The degree of a vertex in a graph indicates how many edges are connected to that vertex. It is an essential concept because:

• Connections: It counts the number of direct connections or neighbors that a particular vertex has.
• Graph Analysis: Knowing the degree can help analyze the overall structure and properties of the graph.

If a vertex has a high degree, it is heavily connected, while a vertex with a low degree has fewer connections. In an r-regular graph, every vertex has the same degree \( r \). This makes it straightforward to calculate structural properties of the graph, since there’s no variance in connection count among vertices.

Sum of Degrees Formula

The sum of degrees formula is a fundamental concept in graph theory. It states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges in the graph. Here's why this works:

• Edge Sharing: Every edge in a graph connects two vertices, so each edge contributes 2 to the sum of degrees (1 for each vertex it connects).

The formula can be expressed as \( ext{Sum of Degrees} = 2 imes ext{Number of Edges} \). This relationship helps in analyzing and solving problems related to graph's edges and vertices, such as determining the number of edges in a regular graph. It's a useful concept for quickly figuring out structural details of any graph setup.

Number of Edges

To determine the number of edges in an r-regular graph, we use the relationship provided by the sum of degrees formula. Here's how:

• Step 1: Calculate the sum of all vertex degrees, which is \( n \times r \) for an r-regular graph with \( n \) vertices.
• Step 2: Set this equal to twice the number of edges: \( n \times r = 2 \times ext{Number of Edges} \).
• Step 3: Solve for the number of edges: \( ext{Number of Edges} = \frac{n \times r}{2} \).

This calculation provides a straightforward way to determine how many edges are in an r-regular graph with a known number of vertices \( n \) and vertex degree \( r \). This formula is especially practical in designing networks and solving graph-based problems.