Problem 36 - Discrete Mathematics with Applications

Question

A simple graph \(G\) is regular if every vertex has the same degree. If every vertex has degree \(r, G\) is \(r\) -regular with \(r\) the degree of the graph. Draw a regular graph with the given properties. \(r=1\) and two vertices.

Step-by-Step Solution

Answer

To draw a simple, regular graph with two vertices (A and B) and degree 1, simply draw two distinct points or circles to represent the vertices and connect them with an edge (a line) to make each vertex have a degree of 1. The graph will look like: \[ A - B \]

Step 1: Draw the vertices

Firstly, draw two distinct vertices (labeled as A and B) on a plane. The vertices should be represented as two distinct points or circles.

Step 2: Connect the vertices

To make the graph regular with r = 1, each vertex must have a degree of 1. This means each vertex must be connected to one other vertex. Draw an edge (a line) between vertices A and B. This will result in both vertices having a degree of 1. After completing these steps, you will have successfully drawn a simple, regular graph with two vertices and degree 1. The graph will look like: ``` A — B ```

Key Concepts

regular graph

In graph theory, a regular graph is a special type of graph where every vertex has the same number of connections or edges. This implies that each point or circle, known as a vertex, "touches" the same number of other vertices.

Imagine a regular graph as a community where each person has an equal number of friendships. Each vertex acts like a person, and each edge connecting vertices represents a friendship. Consequently, regular graphs are balanced or uniform in terms of how each vertex interacts with others.

Regular graphs can vary based on the degree of each vertex, which is a crucial aspect that we'll explore more in the next sections.

vertex degree

The vertex degree in graph theory defines how many connections a vertex has in a graph. Represented simply as the count of edges (or lines) tied to a given vertex, it's a fundamental concept.

For instance, if one vertex has three edges connecting it to other vertices, its degree is 3. Understanding vertex degree is key, as it determines the structure of the graph and affects the ways in which graphs can be categorized, like regular or irregular.

Guided by the assignment above, every vertex in the graph has the same degree, specifically degree 1; this leads us to the concept of regular graphs.

simple graph

A simple graph is one of the most basic types of graphs in graph theory. It’s characterized by its straightforward nature: no loops or multiple edges between the same set of vertices.

The simplicity lies in its connections; each pair of vertices is connected by at most one edge:

• No loop connecting a vertex to itself.
• No double edges connecting the same pair of vertices more than once.

A real-world equivalent might be a transportation map where routes (edges) connect various locations (vertices), with no redundant paths. These properties simplify the study and application of graph theory, making simple graphs foundational in learning.

r-regular graph

An r-regular graph is a specific type of regular graph where each vertex has exactly r connections, or edges. Therefore, it is both regular and customizable, based on the chosen value of r.

If a graph is 2-regular, each vertex is connected to precisely two other vertices, forming shapes like cycles. Our specific example involves an r-regular graph where each vertex connects to just one other vertex, creating a simple line with degree 1.

The notion of r-regular graphs becomes powerful when dealing with complex networks as it ensures uniform connection across all parts of a graph. These graphs are critical in fields like chemistry, where they can model molecules with specific bonding patterns.