Undirected graph

Definition

An undirected graph is a set of vertices connected by edges, where the edges have no direction, meaning the connection between two vertices is mutual. In this type of graph, if vertex A is connected to vertex B, then vertex B is also connected to vertex A. This characteristic plays a significant role in modeling relationships that are inherently bidirectional, such as friendships in social networks or connections in transportation systems.

5 Must Know Facts For Your Next Test

1. In an undirected graph, the adjacency relation between any two vertices is symmetric; if A is adjacent to B, then B is adjacent to A.
2. 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 vertices it connects.
3. Undirected graphs can be represented using adjacency matrices, where a 1 indicates the presence of an edge between two vertices and a 0 indicates absence.
4. The concept of connectivity is crucial in undirected graphs; a graph is connected if there is a path between every pair of vertices.
5. Undirected graphs are widely used in various applications such as network design, clustering algorithms, and social network analysis.

Review Questions

• How does the lack of direction in an undirected graph affect the representation of relationships between vertices?
  • In an undirected graph, the lack of direction means that relationships between vertices are mutual. This characteristic allows for modeling scenarios where interactions are bidirectional, such as friendships where both individuals share the same relationship. Thus, if vertex A is connected to vertex B, it implies that B also recognizes A as connected, effectively creating a symmetric relationship.
• Compare and contrast undirected graphs with directed graphs, particularly regarding their applications in network analysis.
  • Undirected graphs model bidirectional relationships effectively, making them suitable for representing social networks or mutual connections, while directed graphs capture one-way relationships. In network analysis, undirected graphs can simplify scenarios like collaboration networks where each participant has an equal standing. In contrast, directed graphs are more appropriate for situations like web page links or traffic flow where directionality is crucial. The choice between these two types often depends on the specific nature of the relationships being studied.
• Evaluate how the properties of undirected graphs influence algorithms used in graph theory, especially in terms of connectivity and traversal.
  • The properties of undirected graphs significantly impact algorithms such as depth-first search (DFS) and breadth-first search (BFS) for traversing graphs. Since edges are bidirectional, these algorithms can explore all paths without concern for directionality. Additionally, connectivity algorithms utilize the symmetry inherent in undirected graphs to identify components efficiently. Understanding these properties helps optimize algorithm performance and ensures accurate results when analyzing networks and relationships within various data science applications.

Related terms

Directed Graph: A graph where the edges have a direction, indicating a one-way relationship from one vertex to another.

Weighted Graph: A graph in which each edge has an associated weight or cost, often used to represent distances or other quantitative measures.

Subgraph: A portion of a graph formed by a subset of its vertices and the edges connecting them.