Undirected graph

Definition

An undirected graph is a type of graph in which the edges have no direction, meaning that the connection between any two vertices is bidirectional. This allows for a symmetrical relationship between the vertices, where if one vertex is connected to another, it can be traversed in either direction. Undirected graphs are commonly used to represent relationships where the order does not matter, such as social networks or connections in a communication network.

5 Must Know Facts For Your Next Test

1. In an undirected graph, if there is an edge between vertices A and B, it implies a connection that can be traveled from A to B and from B to A.
2. Undirected graphs can be represented visually using points for vertices and lines connecting them for edges, with no arrows indicating direction.
3. The degree of a vertex in an undirected graph is defined as the number of edges connected to it, and each edge contributes to the degree of both its incident vertices.
4. Undirected graphs can contain cycles, where a path starts and ends at the same vertex, forming a closed loop.
5. These graphs can be used to model various real-world systems such as computer networks, transportation systems, and collaboration networks among individuals.

Review Questions

• Compare and contrast undirected graphs with directed graphs regarding their structure and applications.
  • Undirected graphs have edges that indicate a two-way relationship between vertices, whereas directed graphs have edges with direction, showing a one-way relationship. This fundamental difference impacts their applications; undirected graphs are often used in scenarios like social networks where relationships are mutual, while directed graphs are used in situations such as web page links or task scheduling where direction matters. Understanding these differences helps in choosing the right graph type for modeling specific systems.
• How does the concept of vertex degree apply to undirected graphs, and why is it important in analyzing their properties?
  • In undirected graphs, the degree of a vertex is determined by counting how many edges connect to it. This concept is important because it provides insights into the structure of the graph; for example, high-degree vertices can indicate key nodes in a network. Analyzing vertex degrees can help identify important connections and potential bottlenecks within the network, making it a valuable tool in understanding graph behavior.
• Evaluate the role of undirected graphs in real-world applications and discuss how they can influence decision-making processes.
  • Undirected graphs play a significant role in various real-world applications such as social network analysis, where they help visualize connections among individuals. In transportation networks, they assist in finding efficient routes without regard to direction. By using undirected graphs, decision-makers can better understand relationships and interactions within systems, enabling them to make informed choices about resource allocation or strategic planning. The ability to analyze these connections fosters effective communication and collaboration among users.

Related terms

Vertex: A vertex (or node) is a fundamental part of a graph that represents an entity or point in the structure.

Edge: An edge is a connection between two vertices in a graph, representing the relationship between them.

Weighted Graph: A weighted graph is a graph in which edges have associated weights or costs, representing the magnitude of the relationship between connected vertices.