Question

Construct all degree sequences for graphs with four vertices and no isolated vertex.

Short answer

The sequences are (2, 2, 2, 2), (3, 2, 2, 1), and (2, 2, 1, 1).

Step-by-step solution

Understand the Concept of Degree Sequences

A degree sequence is a list of degrees of the vertices in a graph. For a graph with four vertices, each vertex has a degree that is a non-negative integer. The degree of a vertex is the number of edges connected to it. The sequence is often written in non-increasing order.

Interpret No Isolated Vertex Condition

A graph has no isolated vertices if each vertex has a degree of at least 1. Therefore, for a graph with four vertices, each degree in the sequence must be at least 1.

Determine Possible Degree Sum

The sum of degrees in the sequence must satisfy the Handshaking Lemma, which states that it must be even because each edge is incident to two vertices: \[ ext{Sum of Degrees} = 2 imes \text{Number of Edges} \]Since there are no isolated vertices, this sum must be at least 4. The maximum degree of any vertex in a graph with four vertices is 3, so the possible degree sums are 4, 6, and 8.

List Possible Degree Sequences with Sum 4

Only one possible sequence: - (2, 1, 1, 1): This sequence corresponds to a graph where one vertex is connected to all others, and it meets both the sum and non-isolated conditions.

List Possible Degree Sequences with Sum 6

The sequences possible for sum 6 are: - (3, 1, 1, 1): Not possible as one node can't connect to three others with only three potential connections. - (2, 2, 1, 1): Possible. It represents a connected graph like a path of length 3, or a cycle.

List Possible Degree Sequences with Sum 8

The sequences possible for sum 8 are: - (3, 3, 1, 1): Not possible as discussed in Step 5. - (3, 2, 2, 1): Possible, where the first vertex connects to all others and others are connected appropriately. - (2, 2, 2, 2): Also possible, forming a cycle or two disconnected components each with a degree of 2.

Construct All Valid Sequences

Collect all possible sequences from previous steps: - (2, 2, 2, 2) - (3, 2, 2, 1) - (2, 2, 1, 1)

Key concepts

Handshaking Lemma

The Handshaking Lemma is a fundamental concept in graph theory. It basically states that in any graph, the sum of all vertex degrees is twice the number of edges. This is because each edge connects two vertices, effectively being counted twice – once for each vertex it connects.

For any graph, if you add up all the degrees of the vertices, this total sum must be an even number. This makes sense, mathematically, as multiplying two integers (number of edges being the integer here) results in an even number when applied to this formula:

• Sum of Degrees = 2 × Number of Edges

Understanding the Handshaking Lemma is crucial for determining the possible degree sequences of a graph. For instance, in a graph with four vertices, the sum of the degrees must be even, which greatly influences how you can assign degrees to the vertices.

Graph Theory Concepts

Graph theory is a field of mathematics that studies the properties of graphs. A graph in this sense is a collection of points, called vertices, connected by lines, called edges.

One basic understanding in graph theory is the degree of a vertex, which is the number of edges incident to it. In our problem context, we're dealing with degree sequences, which are lists indicating the degree of each vertex in a graph, usually written in non-increasing order. This helps in analyzing the structure of the graph easily.

Furthermore, graph theory concepts also include understanding different types of graphs like paths, cycles, and how these can be structured using corresponding degree sequences. For instance, a path with four vertices might be represented with the degree sequence of (2, 1, 1, 1), indicating how each vertex connects to the others.

Non-Isolated Vertices Condition

The non-isolated vertices condition is an important constraint in creating certain types of graphs. A vertex in a graph is called 'isolated' if it has a degree of zero, meaning no edges are connected to it. If a graph is said to have no isolated vertices, this means that each vertex must have a degree of at least one.

In the problem at hand, this condition dictates that each of the four vertices must connect to at least one other vertex. Therefore, the degree sequence cannot have zeros and must have a sum that aligns with the Handshaking Lemma, emphasizing that the sum should be even.

• The minimum sum is 4, as each vertex has a minimum degree of 1.
• The maximum sum is constrained by the possible highest degree a vertex can have in this context.

This condition effectively helps narrow down the potential valid degree sequences by stating what is not permissible, ensuring each vertex participates in the graph's connections.