Undirected Graph

Definition

An undirected graph is a set of vertices connected by edges, where the edges do not have a direction. This means that if there is an edge between two vertices, it can be traversed in both directions, making it possible to move back and forth between those vertices. Undirected graphs are important because they can represent relationships where the order of connection doesn’t matter, such as friendships or collaborations.

5 Must Know Facts For Your Next Test

1. In an undirected graph, each edge is represented as a pair of vertices, meaning (u, v) is the same as (v, u).
2. An undirected graph can contain loops, which are edges that connect a vertex to itself.
3. A simple undirected graph does not have multiple edges between the same pair of vertices or any loops.
4. Undirected graphs can be represented using adjacency lists or edge lists, which help in efficient storage and traversal.
5. The maximum number of edges in a simple undirected graph with n vertices is given by the formula $$\frac{n(n-1)}{2}$$.

Review Questions

• Compare and contrast undirected graphs with directed graphs in terms of connectivity and traversal.
  • In an undirected graph, edges have no direction, allowing traversal between connected vertices in both directions. This means if vertex A is connected to vertex B, you can go from A to B and also from B back to A freely. In contrast, directed graphs have edges that indicate a one-way relationship, meaning you can only traverse from one vertex to another in the specified direction. This difference affects how paths and cycles are analyzed within the two types of graphs.
• Evaluate how the degree of a vertex in an undirected graph can influence the overall structure and connectivity of the graph.
  • The degree of a vertex in an undirected graph significantly affects its connectivity and the overall structure. Vertices with high degrees serve as critical hubs that connect many other vertices, enhancing connectivity and potentially making the graph more robust. Conversely, vertices with low degrees may indicate areas of vulnerability or isolation within the graph. Understanding the degree distribution helps analyze properties like network resilience and the potential for information flow across the entire graph.
• Assess how different representations of undirected graphs (such as adjacency lists and edge lists) can impact computational efficiency for various algorithms.
  • Different representations of undirected graphs significantly affect computational efficiency when performing algorithms. Adjacency lists provide quick access to neighboring vertices, making them ideal for sparse graphs where many vertices do not have edges. On the other hand, edge lists are simple to implement but may require more time for searching through edges when checking for connectivity or performing traversal algorithms. The choice of representation thus influences algorithm complexity and performance depending on the specific use case and properties of the graph.

Related terms

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

Edge:

An edge is a connection between two vertices in an undirected graph, indicating a relationship without any direction.

Degree:

The degree of a vertex in an undirected graph is the number of edges connected to it, reflecting how many connections it has with other vertices.