Question

Determine all possible degree sequences for graphs with five vertices containing no isolated vertex and six edges.

Short answer

Possible degree sequences: [4, 4, 2, 1, 1], [4, 3, 2, 2, 1], [3, 3, 3, 2, 1].

Step-by-step solution

Understand Degree Sequences

A degree sequence is a sequence of vertex degrees of a graph, typically arranged in non-increasing order. We need to consider all possible degree sequences for a graph with 5 vertices and no isolated vertices, totaling 6 edges.

Calculate Total Degree Sum

For any graph with 5 vertices and 6 edges, the sum of degrees equals twice the number of edges, due to the handshaking theorem. Thus, for 6 edges, the total degree sum is: \[2 \times 6 = 12.\]

Consider Non-Isolated Constraint

Since the graph has no isolated vertices, each vertex must have a degree of at least 1.

List Possible Sequences

Enumerate all sequences of 5 numbers that add to 12 (the degree sum), with each number at least 1. Each sequence must represent possible vertex degrees without exceeding the number of vertices minus one (maximum degree is 4).

Verify Degree Sequence Validity

Check possible sequences against graph theory constraints, including:\( [4, 4, 2, 1, 1]\), \( [4, 3, 2, 2, 1]\), and \( [3, 3, 3, 2, 1]\). Ensure they follow the handshaking lemma and form valid graphs.

Verify with Examples

Each valid sequence leads to different graph structures. Visualizing or constructing graphs helps ensure that the degree sequences are possible while maintaining connectivity and edge counts.

Key concepts

Non-Isolated Vertices

In graph theory, a non-isolated vertex is a fundamental concept. When we say a vertex is non-isolated, we mean it must have at least one edge connected to it. This is crucial because isolated vertices stand alone without any connections, which doesn't meet the criteria when working with graphs where all vertices must interact.

Let's break it down further:

• In graphs, the vertices are points that may or may not be connected by edges.
• A non-isolated vertex has connections, implying it is not a stand-alone point.
• For a graph with five vertices and no isolated vertices, each vertex must have a minimum degree of 1.

In the context of our problem, ensuring no isolated vertices means all vertices are part of the network, contributing to the overall degree sum discussed next.

Handshaking Theorem

The handshaking theorem is a guiding principle in graph theory that helps us determine relationships between vertices and edges in a graph. According to this theorem, the sum of the degrees of all vertices in a finite graph is equal to twice the number of edges.

This theorem is particularly helpful because it allows us to check the consistency of a graph and ensure our calculations are correct. For a graph with six edges, the handshaking theorem tells us that:

• The total sum of degrees should be \(2 \times 6 = 12\).
• Each edge contributes to two vertex degrees (one for each endpoint).

Using the theorem, you can easily figure out if your graph’s degree sequence is plausible by seeing if the total degree sum matches what's expected according to the number of edges. This helps ensure the graph can actually exist with the given parameters.

Graph Theory Constraints

Graph theory constraints are rules that govern the possible structures and behaviors of graphs. When you discover potential degree sequences, these constraints are your guide for determining their validity.

Let's explore these constraints:

• The sum of degrees must align with the handshaking theorem, ensuring each sequence reflects a plausible graph.
• Each vertex's degree can't exceed the total number of vertices minus one (for five vertices, the max degree is 4).
• In the problem, when considering graphs with non-isolated vertices, each degree must be at least 1.

These constraints help exclude impossible configurations and guide the enumeration of feasible degree sequences. For instance, sequences like \([4, 4, 2, 1, 1]\), while meeting the sum constraint, must still be checked to confirm connectivity and adherence to vertex degree limits within a practical graph structure.