Degree

Definition

In graph theory, the degree of a vertex is defined as the number of edges incident to that vertex. This concept is essential for understanding connectivity within a graph, as it helps describe how many connections each vertex has, which can indicate how well a vertex can reach other vertices. The degree also plays a crucial role in various properties and behaviors of graphs, such as Eulerian paths and circuits, which depend on the degree of vertices.

5 Must Know Facts For Your Next Test

1. The degree of a vertex can be classified as either 'in-degree' (number of incoming edges) or 'out-degree' (number of outgoing edges) in directed graphs.
2. In an undirected graph, the sum of the degrees of all vertices equals twice the number of edges in the graph.
3. A vertex with a degree of zero is known as an isolated vertex because it has no connections to any other vertex.
4. Graphs with all vertices having even degrees are essential for the existence of Eulerian circuits, while those with exactly two vertices of odd degree allow for Eulerian paths.
5. The concept of degree helps determine the connectivity and overall structure of a graph, influencing algorithms for traversal and searching.

Review Questions

• How does the degree of a vertex affect its connectivity within a graph?
  • The degree of a vertex directly impacts its connectivity since a higher degree means more edges are connected to that vertex. This allows for greater accessibility to other vertices in the graph. For instance, if a vertex has many connections, it can serve as a hub for traversing the graph, whereas a low-degree vertex may limit pathways to reach other parts of the graph.
• What implications does the degree of vertices have on finding Eulerian paths and circuits in a graph?
  • The degree of vertices plays a significant role in determining the presence of Eulerian paths and circuits. For an Eulerian circuit to exist, every vertex must have an even degree. Conversely, if exactly two vertices have an odd degree, then an Eulerian path exists. Understanding these degree conditions helps in identifying whether certain types of traversals are possible in the graph.
• Evaluate how changes in vertex degrees within a network can impact its overall functionality and robustness.
  • Changes in vertex degrees can significantly affect a network's functionality and robustness. For example, if key vertices lose connections (decreasing their degree), it could create bottlenecks or isolated sections within the network. Such alterations may reduce overall efficiency and resilience to failures, highlighting how crucial high-degree vertices are for maintaining connectivity and enabling efficient communication across the network.

Related terms

vertex: A fundamental unit of a graph that represents an endpoint or node where edges meet.

edge: A connection between two vertices in a graph, representing a relationship or pathway.

connected graph: A type of graph in which there is a path between every pair of vertices, ensuring that the entire graph is accessible.