Undirected graph

Definition

An undirected graph is a collection of nodes connected by edges, where the edges do not have a direction. This means that if there is an edge between two nodes, it can be traversed in both directions equally. The absence of direction allows for simpler relationships between nodes, making undirected graphs particularly useful in modeling scenarios where mutual connections exist, such as social networks or transportation systems.

5 Must Know Facts For Your Next Test

1. In an undirected graph, the relationship between any two connected vertices is symmetric; if vertex A is connected to vertex B, then B is also connected to A.
2. Undirected graphs can be represented using adjacency lists or adjacency matrices, allowing for efficient storage and manipulation of the graph structure.
3. They are widely used in various applications such as computer networks, social networks, and graph-based algorithms like depth-first search and breadth-first search.
4. A complete undirected graph is one in which every pair of distinct vertices is connected by a unique edge, leading to a highly interconnected structure.
5. Cycles in undirected graphs are closed paths where you can start at a vertex and return to it without retracing any edges.

Review Questions

• How does the absence of direction in an undirected graph impact the relationships between its vertices?
  • The absence of direction in an undirected graph means that relationships between vertices are bidirectional. If one vertex is connected to another, it indicates a mutual connection, allowing traversal from either vertex to the other without restriction. This property simplifies many algorithms and applications by treating connections as inherently symmetric.
• Compare and contrast the representation of undirected graphs using adjacency lists versus adjacency matrices.
  • Adjacency lists represent undirected graphs by storing a list of neighbors for each vertex, which can save space for sparse graphs. In contrast, adjacency matrices use a two-dimensional array to represent connections between all pairs of vertices, making it easier to check if an edge exists but potentially wasting space for sparse graphs. Each representation has its strengths depending on the specific application and the density of edges.
• Evaluate the role of undirected graphs in real-world applications and how their properties facilitate complex problem-solving.
  • Undirected graphs play a significant role in real-world applications such as modeling social networks where relationships are inherently mutual or designing transportation systems that connect multiple locations. Their properties enable complex problem-solving through algorithms that leverage connectivity, such as finding shortest paths or detecting clusters. The ability to analyze these graphs helps optimize resources and improve efficiency in various fields like logistics and network design.

Related terms

vertex: A vertex is a fundamental unit of an undirected graph, representing a point where edges meet.

edge: An edge is a line that connects two vertices in an undirected graph, indicating a relationship or connection between them.

degree: The degree of a vertex in an undirected graph is the number of edges connected to it, reflecting its connectivity within the graph.