Determining the Maximum and Minimum Number of Vertices in a Simple Graph with 16 Edges

When working with simple graphs where each vertex has the same degree and the minimum degree is at least 4, it's crucial to understand the relationship between the number of edges, vertices, and the degree of each vertex. This article will explore the problem of finding the maximum and minimum number of vertices in such a graph, and provide a simple example to clarify the concept.

Understanding the Basic Concepts

In graph theory, a simple graph is defined as a graph that does not contain loops (edges from a vertex to itself) or multiple edges between the same pair of vertices. Given a simple graph with V vertices and E edges, the degree of each vertex is the number of edges incident to that vertex.

Key Relationship in Simple Graphs

The Handshake Lemma states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges. Mathematically, this can be expressed as:

S 2E

Achieving a Common Degree of 4

Given that the degree d of each vertex must be at least 4, we can express the degree sum as:

4V 2E

Furthermore, we know that the graph has 16 edges, so:

2E 32

Substituting this into the equation for the degree sum gives:

4V 32

From this equation, we can solve for the number of vertices V in the graph:

V 32 / 4 8

Hence, the simple graph has a maximum of 8 vertices if each vertex has a degree of 4.

Minimum Number of Vertices

To find the minimum number of vertices, we need to consider the minimum degree, which is 4. Using the same relationship, we can express the minimum number of vertices as:

4V 32

When each vertex has a degree of 16 (the maximum possible degree given 16 edges), we get:

V 32 / 16 2

This means that the minimum number of vertices, where each vertex has a degree of 16, is 2. However, this scenario is not feasible because the maximum degree of a vertex in a simple graph with 2 vertices is 1, which is less than 4.

Example: Constructing a Graph with 8 Vertices

To construct a simple graph with 16 edges where each vertex has a degree of 4, we can use the example of a cube. A cube has 8 vertices and 12 edges, with each vertex having a degree of 3. We can modify this to create a graph with the required properties:

Split the 8 vertices of the cube into 4 pairs of non-adjacent vertices (for example, the face diagonals). Use the remaining 4 out of 16 edges to join these pairs of vertices.

An even simpler example is to consider the complete graph on 4 vertices (K4). This graph has 6 edges and each vertex has a degree of 3. By making two copies of K4 and joining them with 4 additional edges, we can create a graph with 16 edges and each vertex having a degree of 4.

By understanding these relationships and applying them, we can determine the number of vertices in a simple graph based on the given conditions. This knowledge is invaluable for students and professionals in graph theory and related fields, helping to optimize and analyze various network configurations more effectively.