Degree Sequence

Definition

A degree sequence is a list of the degrees of the vertices in a graph, usually arranged in non-increasing order. It provides a way to summarize the connectivity of a graph and is crucial for understanding various properties, such as whether a particular graph can exist or its structure. The degree sequence helps in applying key concepts like the Handshaking Lemma, which relates the sum of the degrees to the number of edges, and in identifying special types of graphs such as bipartite, complete, and regular graphs.

5 Must Know Facts For Your Next Test

1. A degree sequence can determine whether a simple graph is possible through the Havel-Hakimi algorithm, which provides a method for testing graphical sequences.
2. The degree sequence of a bipartite graph must consist of two distinct sets of degrees that correspond to each partition of vertices.
3. In a complete graph with n vertices, the degree sequence consists of n-1 repeated n times since every vertex is connected to all other vertices.
4. A regular graph's degree sequence will show identical entries, indicating that all vertices are equally connected.
5. The sum of the degrees in any graph must be even because it equals twice the number of edges, as stated by the Handshaking Lemma.

Review Questions

• How does the degree sequence relate to the Handshaking Lemma and what implications does this have for understanding graph properties?
  • The degree sequence directly relates to the Handshaking Lemma, which states that the sum of all vertex degrees in a graph equals twice the number of edges. This implies that if you sum up the degree sequence and it results in an odd number, then such a graph cannot exist since it contradicts the lemma. Understanding this connection helps us quickly evaluate potential graphs and their connectivity.
• Explain how the degree sequence can help distinguish between different types of graphs, particularly bipartite and regular graphs.
  • The degree sequence aids in identifying different types of graphs by analyzing their patterns. For bipartite graphs, you will see two distinct sets in the degree sequence representing each partition, while for regular graphs, all entries will be identical since each vertex shares the same number of connections. This distinction allows us to classify and understand graph structures more easily.
• Evaluate how knowledge of degree sequences impacts our understanding of network design and analysis in real-world applications.
  • Knowledge of degree sequences is crucial in network design as it helps predict how changes in connectivity can affect overall performance. For instance, by analyzing degree sequences, we can identify bottlenecks or critical nodes within communication networks or social networks. A well-understood degree sequence allows for optimizing connections to ensure robust and efficient network operation, making it valuable in fields like telecommunications and transportation planning.

Related terms

Handshaking Lemma:

A fundamental principle stating that in any graph, the sum of all vertex degrees is equal to twice the number of edges.

Bipartite Graph:

A type of graph whose vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other.

Regular Graph:

A graph where every vertex has the same degree, meaning the number of edges connected to each vertex is constant.