Graphical Sequences

Definition

A graphical sequence is a list of non-negative integers that can represent the degree sequence of a simple graph, meaning it can be realized as a graph with those degrees for each vertex. This concept is crucial when discussing different types of graphs, such as bipartite, complete, and regular graphs, as it helps determine the existence of such graphs based on their degree sequences. By analyzing whether a sequence is graphical, one can understand the structural properties and characteristics of the corresponding graph.

5 Must Know Facts For Your Next Test

1. A sequence is graphical if it satisfies the Handshaking Lemma, which states that the sum of the degrees must be even since each edge contributes to two vertices' degrees.
2. The Havel-Hakimi algorithm iteratively reduces the degree sequence by removing the largest degree and reducing the next largest degrees accordingly, checking if it remains graphical.
3. Complete graphs have a degree sequence where all vertices have degrees equal to the total number of vertices minus one.
4. Regular graphs are characterized by having all vertices with the same degree, leading to unique graphical sequences based on their regularity.
5. In bipartite graphs, the degree sequences of both sets can influence the overall graphical nature and must adhere to specific properties based on how many connections can exist between the two sets.

Review Questions

• How does the Handshaking Lemma relate to determining whether a sequence is graphical?
  • The Handshaking Lemma states that in any graph, the sum of the degrees of all vertices must be even since each edge contributes to the degrees of two vertices. Therefore, for a sequence to be graphical, its total must also be even. This serves as a necessary condition for any sequence to represent a valid degree sequence for a simple graph, linking directly to the structural properties needed for creating such graphs.
• Discuss how the Havel-Hakimi algorithm can be utilized to test if a degree sequence corresponds to a bipartite or regular graph.
  • The Havel-Hakimi algorithm systematically checks if a given degree sequence can form a valid graph by iteratively reducing it. For bipartite graphs, it’s essential that after applying the algorithm, the resulting sequences match specific conditions related to their partitioning into two sets. Similarly, for regular graphs, if all remaining degrees are identical after processing through Havel-Hakimi, it indicates that a regular graph can be constructed from that sequence.
• Evaluate how understanding graphical sequences enhances our comprehension of special types of graphs like complete and bipartite graphs.
  • Understanding graphical sequences provides insight into how different types of graphs are formed and their inherent properties. For example, complete graphs have degree sequences where each vertex connects to every other vertex, while bipartite graphs require that connections only occur between two distinct sets. By analyzing graphical sequences, we can discern whether specific configurations can exist and how they relate structurally, thus enhancing our overall comprehension of graph theory.

Related terms

Degree Sequence:

A degree sequence is an ordered list of the degrees of the vertices in a graph, reflecting how many edges are incident to each vertex.

Havel-Hakimi Algorithm:

The Havel-Hakimi algorithm is a method used to determine whether a given sequence of non-negative integers can be the degree sequence of a simple graph.

Bipartite Graph:

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