Undirected graph

Definition

An undirected graph is a type of graph in which the edges have no direction, meaning that the connections between the vertices (or nodes) are bidirectional. This allows for a symmetric relationship between connected vertices, where the order of the vertices does not affect the nature of the connection. Undirected graphs are commonly used to represent relationships where direction is not important, such as friendship networks or collaborations.

5 Must Know Facts For Your Next Test

1. In an undirected graph, if there is an edge connecting vertex A to vertex B, it implies a two-way connection, meaning A is connected to B and B is connected to A.
2. The degree of a vertex in an undirected graph is the number of edges connected to it, providing insight into its connectivity within the network.
3. Undirected graphs can be represented visually using circles for vertices and lines for edges, making it easier to interpret relationships and structures.
4. One common application of undirected graphs is in social network analysis, where individuals are represented as vertices and friendships as edges.
5. Undirected graphs can also be used to model pathways or networks in biological systems, such as metabolic pathways or protein interactions.

Review Questions

• Compare and contrast undirected graphs with directed graphs, focusing on their structural differences and implications for representing relationships.
  • Undirected graphs differ from directed graphs in that their edges do not have a specified direction, indicating a bidirectional relationship between vertices. In contrast, directed graphs have edges that point from one vertex to another, representing one-way relationships. This distinction is crucial when modeling systems; for example, undirected graphs are ideal for scenarios like social networks where mutual connections exist, while directed graphs are used for scenarios with asymmetric relationships such as web page links.
• Discuss how the concept of degree in an undirected graph helps analyze connectivity and significance of individual vertices within a network.
  • The degree of a vertex in an undirected graph represents the number of edges connected to it, serving as a key metric for assessing its importance within the network. Higher degree vertices often indicate more influential or well-connected entities. By analyzing vertex degrees across the network, patterns of connectivity emerge, highlighting central nodes that may play crucial roles in information flow or interaction dynamics. This analysis can inform strategies in fields like epidemiology or marketing.
• Evaluate the applications of undirected graphs in modeling real-world systems and how they enhance our understanding of complex networks.
  • Undirected graphs are widely utilized in various fields to model complex networks, offering valuable insights into relational dynamics. For instance, in biology, they can represent metabolic pathways where reactions are bidirectional. In social sciences, they capture mutual friendships in social networks. By employing algorithms on these graphs, researchers can uncover community structures and analyze network resilience. This enhances our understanding by revealing how changes within one part of the network can influence overall behavior and stability.

Related terms

Vertex: A vertex, also known as a node, is a fundamental part of a graph that represents an entity or point in the network.

Edge:

An edge is a connection between two vertices in a graph, representing the relationship or link between them.

Weighted graph: A weighted graph is a graph in which each edge has an associated weight or cost, often used to represent distances or values between connected vertices.