Undirected graph

Definition

An undirected graph is a collection of nodes (or vertices) connected by edges, where the connections do not have a specific direction. This means that if there is an edge between two nodes, you can travel in either direction between them, representing a symmetric relationship. This property makes undirected graphs suitable for modeling situations where the relationship between entities is bidirectional, such as friendships in social networks or routes in transportation systems.

5 Must Know Facts For Your Next Test

1. In an undirected graph, the degree of a vertex is defined as the number of edges connected to it, reflecting how many direct relationships it has.
2. Undirected graphs can be used to represent various real-world scenarios, such as social networks where friendships are mutual.
3. The representation of an undirected graph can be done using an adjacency matrix or an adjacency list, both of which efficiently depict the connections between vertices.
4. An undirected graph can contain cycles, which are paths that start and end at the same vertex without traversing any edge more than once.
5. When analyzing undirected graphs, concepts like connectivity and components are crucial to understanding how groups of nodes relate to each other.

Review Questions

• How does the absence of direction in edges affect the properties of an undirected graph compared to a directed graph?
  • In an undirected graph, edges imply a two-way connection between nodes, meaning that if there is an edge between node A and node B, you can traverse from A to B and from B to A equally. This contrasts with directed graphs, where each edge has a specified direction. As a result, properties like connectivity and pathfinding become simpler in undirected graphs since all relationships are symmetric.
• Discuss how an undirected graph can be represented and the advantages of each representation method.
  • An undirected graph can be represented using either an adjacency matrix or an adjacency list. The adjacency matrix provides a quick way to check if there is a connection between any two vertices by using indices, but it can be memory-intensive for large graphs with many vertices and few edges. On the other hand, the adjacency list is more space-efficient because it only stores edges for existing connections, making it better for sparse graphs. Each method has its advantages depending on the specific use case and size of the graph.
• Evaluate the role of cycles in undirected graphs and their significance in real-world applications.
  • Cycles in undirected graphs indicate paths that revisit nodes, which can represent recurring relationships or closed-loop systems in real-world scenarios. For instance, in transportation networks, cycles might reflect routes that allow vehicles to return to their starting points efficiently. Understanding cycles helps in optimizing routes and resource allocation in applications such as logistics and network design, revealing how entities interact over repeated engagements.

Related terms

Directed Graph: A directed graph is a set of nodes connected by edges that have a specific direction, indicating a one-way relationship between the nodes.

Vertex: A vertex (or node) is a fundamental unit of a graph, representing an entity or point in the network.

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