Problem 17 - Discrete Mathematics with Applications

Question

Exactly five vertices with degrees \(1,1,1,1,\) and \(4\).

Step-by-Step Solution

Answer

The given vertex degrees are 1, 1, 1, 1, and 4. According to the Handshaking Lemma, the sum of all vertex degrees must be even for the graph to exist. In this case, the sum is 8, which is even, so a graph can exist. Labeling the vertices A, B, C, D, and E, we connect vertices A, B, C, and D to E, satisfying the given vertex degrees. Therefore, the graph exists and follows the given constraints.

Step 1: Verify if the graph can exist

According to the Handshaking Lemma, the sum of degrees of all vertices in a graph must be even since each edge contributes to the degree of two vertices. So, we sum the given degrees: \[ \text{Total degrees} = 1+1+1+1+4 = 8 \] Since 8 is even, the Handshaking Lemma is satisfied, which means a graph with these degrees can exist.

Step 2: Create the graph

Let's label the vertices as A, B, C, D, and E, in such a way that Vertices A, B, C, and D have a degree of 1, and Vertex E has a degree of 4. Now we will connect the vertices based on their degrees. 1. Connect Vertex A to Vertex E (degree of Vertex A becomes 1, degree of Vertex E becomes 1) 2. Connect Vertex B to Vertex E (degree of Vertex B becomes 1, degree of Vertex E becomes 2) 3. Connect Vertex C to Vertex E (degree of Vertex C becomes 1, degree of Vertex E becomes 3) 4. Connect Vertex D to Vertex E (degree of Vertex D becomes 1, degree of Vertex E becomes 4)

Step 3: Verify the result

Now, we have a graph connecting vertices A, B, C, and D to vertex E. We can observe that the degrees of every vertex match the given problem. Vertex A, B, C, and D have a degree of 1, and Vertex E has a degree of 4. Thus, such a graph exists and follows the given constraints.

Key Concepts

Handshaking Lemma

The Handshaking Lemma is a fundamental concept in graph theory that states: the sum of all vertex degrees in an undirected graph must be even. This is because each edge connects two vertices, contributing to two degrees in total. Think of it as shaking hands at a party; every handshake involves two people.

For example, if we count all handshakes (or edges) and multiply by two, we get the sum of all degrees. In graph problems like the exercise given, using the Handshaking Lemma helps determine whether it is possible to construct such a graph. If the sum of the degrees in the list is not even, the graph cannot exist.

In the example, the degrees are \(1, 1, 1, 1, 4\), totaling \(8\), which is even; hence, a graph can exist.

Vertex Degree

The degree of a vertex in a graph is simply the number of edges connected to it. It can also be understood as how many connections, or 'handshakes,' it has. Degrees can tell us a lot about a graph's structure.

In the exercise, we found five vertices with degrees \(1,1,1,1,\) and \(4\). Vertices with a degree of 1 are often referred to as leaf nodes. These connect only to one other vertex.

The vertex with a degree of 4 is more connected than the others, meaning it serves as a hub connecting to multiple vertices. Understanding vertex degrees is crucial for tasks like graph construction and analysis, allowing you to visualize connections better.

Graph Construction

Constructing a graph from vertex degrees involves making connections (edges) among vertices according to their degrees. This is akin to completing a puzzle—each piece (vertex) must perfectly fit with others to maintain the specified connections.

For our exercise:

• Label vertices \(A, B, C, D, E\). Vertices \(A, B, C, D\) each need one connection.
• Vertex \(E\) requires four connections, serving as the central hub.

The process involves connecting each lesser-degree vertex (\(A,B,C,D\)) exclusively to \(E\). Thus, \(E\) reaches its degree of 4 while \(A, B, C, D\) each reach their degree of 1.

Successful graph construction logically follows the list of degrees and verifies connections adhere to those rules.

Graph Verification

Graph verification ensures if the constructed graph matches the initial problem's conditions and constraints. It requires checking that each vertex's degree is as specified in the list provided.

After constructing the graph, verify by recounting each vertex's degree:

• Vertices \(A, B, C,\) and \(D\) connect once each—degree 1.
• Vertex \(E\) connects four times, once to each of the others—degree 4.

This process confirms that the graph accurately represents the intended degrees from the start of the problem, thereby verifying that the graph solution is valid. By following this step, one ensures all connections are correct and satisfy the initial problem's requirements.

Verification is a crucial step, safeguarding against errors and ensuring the accuracy of the constructed graph.