Undirected graph

Definition

An undirected graph is a set of objects (vertices or nodes) connected by edges that have no direction. This means that the relationship between any two connected vertices is bidirectional, indicating that if vertex A is connected to vertex B, then vertex B is also connected to vertex A. Undirected graphs are fundamental in various applications, as they can represent symmetric relationships such as friendships in social networks or roads connecting cities.

5 Must Know Facts For Your Next Test

1. In an undirected graph, if there is an edge between vertex A and vertex B, it implies a two-way connection, so there is no concept of 'starting' or 'ending' at either vertex.
2. Undirected graphs can be represented using adjacency matrices where the entry at row i and column j indicates whether there is an edge between vertices i and j.
3. The number of edges in an undirected graph can vary significantly based on the number of vertices and the type of relationships being represented.
4. Undirected graphs can also contain cycles, which are paths that start and end at the same vertex without traversing any edge more than once.
5. Common algorithms like Depth-First Search (DFS) and Breadth-First Search (BFS) can be applied to undirected graphs to traverse or explore their structure.

Review Questions

• How does the bidirectional nature of edges in an undirected graph influence the way we interpret connections between vertices?
  • The bidirectional nature of edges in an undirected graph means that each connection between vertices represents a mutual relationship. This interpretation allows us to model scenarios where the relationship is symmetric, such as friendships where both individuals acknowledge each other. Consequently, this impacts how we analyze connectivity and flow within networks, since every edge effectively doubles the potential paths between vertices.
• Discuss how adjacency matrices can be used to represent undirected graphs and what properties arise from this representation.
  • Adjacency matrices provide a structured way to represent undirected graphs by using a square matrix where rows and columns correspond to vertices. If there is an edge between two vertices, the corresponding matrix entry is marked with a 1; otherwise, it's 0. Since edges are bidirectional, the matrix is symmetric about its diagonal, meaning that the entry for vertex A to vertex B is equal to that from B to A. This symmetry aids in identifying connectivity patterns quickly.
• Evaluate the significance of degrees in undirected graphs and how they can affect graph algorithms like traversal methods.
  • The degree of a vertex in an undirected graph plays a critical role in understanding its structure and connectivity. High-degree vertices often serve as hubs in networks, influencing traversal algorithms like DFS or BFS by prioritizing paths through these well-connected nodes. Furthermore, knowing the degree distribution helps identify characteristics such as centrality or clusters within the graph, guiding more effective strategies for exploration and analysis.

Related terms

Vertex: A vertex is a fundamental unit of an undirected graph, representing a point where edges meet. In simpler terms, it can be thought of as a 'node' in the graph.

Edge: An edge in an undirected graph represents the connection between two vertices, showcasing the relationship without any direction. Each edge connects exactly two vertices.

Degree: The degree of a vertex in an undirected graph refers to the number of edges that are incident to that vertex, indicating how many connections it has to other vertices.