Problem 46 - Discrete Mathematics with Applications

Question

Can there be a 1-regular graph with three vertices?

Step-by-Step Solution

Answer

No, it is not possible to have a 1-regular graph with three vertices. Considering a graph with vertices A, B, and C, if vertex A is connected to vertex B, then vertex B already fulfills the 1-regular condition. Connecting vertex B to vertex C would violate this condition. Consequently, vertex C is left without a connection, making it impossible to construct a 1-regular graph with three vertices.

Step 1: Recalling the definition of a 1-regular graph

A 1-regular graph is a graph in which every vertex has a degree of 1, meaning each vertex is connected to exactly one other vertex by an edge.

Step 2: Three vertices and the connections

Now let's consider a graph with 3 vertices, A, B, and C. Following the 1-regular graph's property, vertex A must be connected to one and only one other vertex. Let's say it is connected to vertex B.

Step 3: Exploring the possibilities for Vertex B

At this point, vertex A is connected to vertex B. Since A is connected to B, vertex B already has a degree of 1, which fulfills the 1-regular graph requirement. Now, we cannot connect vertex B to any other vertex (such as vertex C), as this would increase its degree to 2, violating the condition of being 1-regular.

Step 4: Considering Vertex C

As we have seen, vertex B is already connected to vertex A, and we cannot connect it to another vertex. Thus, vertex C is left without a connection, making it impossible to construct a 1-regular graph with three vertices.

Step 5: Conclusion

It is not possible to have a 1-regular graph with three vertices, as we cannot fulfill the requirement of each vertex having a degree of 1 without leaving one vertex unconnected.

Key Concepts

Degree of a Vertex

In graph theory, the degree of a vertex is a fundamental concept. It denotes the number of edges incident to the vertex. In simple terms, it shows how many connections a vertex has.
An edge is a connection between two vertices, so the more edges a vertex has, the higher its degree.

• A vertex with no edges is said to have a degree of 0.
• If a vertex connects to one other vertex, it has a degree of 1.
• For vertices with multiple connections, the degree equals the count of these connections.

Understanding the degree of vertices helps us explore various properties and types of graphs. Degrees can vary greatly depending on the structure and rules of the graph at hand.

Graph Theory

Graph theory is a branch of mathematics focused on studying graphs. A graph is a collection of points, called vertices, connected by lines, known as edges.
This field explores how vertices are linked and the properties that arise from these connections. Graph theory has numerous applications:

• In computer science, to model networks like social media or communication networks.
• In biology, to understand biological pathways and ecosystems.
• In logistics, to optimize routes and networks.

The study of graph theory involves investigating different types of graphs, such as regular graphs, directed graphs, and weighted graphs, each with unique properties and uses.

Regular Graph Concept

A regular graph is a type of graph where each vertex has the same degree. This means that every vertex is equally connected to the same number of vertices.
A regular graph can be further categorized based on the degree of its vertices. There are different types:

• 1-regular graph: Each vertex has exactly one connection. This typically forms a set of disconnected pairs (often called a perfect matching).
• 2-regular graph: Every vertex connects to two others, often forming simple cycles or loops.

The concept simplifies analyzing and constructing graphs by establishing uniform connectivity. Regular graphs serve as important models in various fields of science and engineering, allowing for consistency in network analysis.