Question

If \(G\) has \(n\) vertices and is regular of degree \(r\), how many edges has \(G\) ? Use your answer to check the number of edges in the Petersen graph and the \(k\)-cube \(Q_{k}\).

Short answer

The number of edges in a regular graph is \(\frac{n \cdot r}{2}\), where \(n\) is the number of vertices and \(r\) is the degree. Specifically, the Petersen graph has 15 edges and a \(k\)-cube has \(k \cdot 2^{k-1}\) edges.

Step-by-step solution

Compute number of edges in general graph \(G\)

We multiply the number of vertices \(n\) by the degree \(r\), then divide by 2 to account for double-counting, i.e., the number of edges in \(G\) is given by \(E = \frac{n \cdot r}{2}\).

Apply to Petersen graph

The Petersen graph has 10 vertices and each vertex is of degree 3. Therefore we can substitute this into the formula to get \(E = \frac{10 \cdot 3}{2} = 15\). Hence the Petersen graph has 15 edges.

Apply to \(k\)-cube graph \(Q_{k}\)

A \(k\)-cube has \(2^{k}\) vertices and each vertex is of degree \(k\). Substituting these values into the formulas gives \(E = \frac{2^{k} \cdot k}{2} = k \cdot 2^{k-1}\). So a \(k\)-cube has \(k \cdot 2^{k-1}\) edges.

Key concepts

Understanding Regular Graphs

A regular graph is a type of graph where every vertex has the same number of neighbors; this number is called the degree of the vertices. When a graph is labeled as \(r\)-regular, it indicates that each vertex in the graph has a degree \(r\).
Regular graphs are a significant concept in graph theory because they exhibit symmetrical properties. These properties allow for straightforward computation of graph characteristics, such as the number of edges.

• In an \(n\)-vertex \(r\)-regular graph, each vertex connects to exactly \(r\) other vertices.
• To find the total number of edges in such a graph, you calculate \(E = \frac{n \cdot r}{2}\). This formula ensures you don't double-count the connections between vertices.

Understanding regular graphs is fundamental before exploring more complex ones like the Petersen graph and \(k\)-cubes.

Exploring the Petersen Graph

The Petersen graph is one of the most famous graphs in the field due to its unique properties and structure. It is a specific type of regular graph with some interesting features.
The Petersen graph consists of 10 vertices, each with a degree of 3, making it a 3-regular graph.

• The symmetrical and highly connected nature of this graph makes it useful in theoretical applications such as demonstrating a counterexample in graph theory problems.
• Using the regular graph edge formula \(E = \frac{n \cdot r}{2}\), we find that the Petersen graph has \(10 \) vertices and \(15\) edges, reinforcing its regularity.

This graph often serves as a textbook example in graph theory due to its counterintuitive properties and visual complexity.

Understanding the k-cube Graph

A \(k\)-cube, also known as a hypercube, is an extension of a square or cube into \(k\) dimensions. It has interesting properties that grow in complexity as \(k\) increases.
Each vertex in a \(k\)-cube is connected to \(k\) other vertices. However, the number of vertices doubles with each added dimension.

• A \(k\)-cube contains \(2^k\) vertices.
• Each vertex shares an edge with \(k\) other vertices, making it a \(k\)-regular graph.
• To compute the edges, apply the formula \(E = k \cdot 2^{k-1}\), which shows how the structure scales with higher dimensions.

The \(k\)-cube is an essential concept when studying network structures and parallel processing systems.

Grasping the Degree of a Vertex

In graph theory, understanding the degree of a vertex is fundamental. The degree tells you how many edges connect to a particular vertex.

• In a simple undirected graph, the degree is the total number of direct connections or edges a vertex has.
• This concept extends to regular graphs, where all vertices share the same degree.

The vertex degree plays a vital role in identifying the type of graph and its properties, like the total number of edges or its regularity.
Recognizing the degree of a vertex helps in determining the graph's structure and is a stepping stone for delving deeper into more advanced graph theory concepts.