An undirected graph is a collection of vertices connected by edges where the edges have no direction. This means that the connection between any two vertices is bidirectional, allowing movement from one vertex to another without a specified starting or ending point. This characteristic enables undirected graphs to represent relationships where the direction does not matter, such as friendships in social networks or connections in computer networks.
congrats on reading the definition of undirected graph. now let's actually learn it.
In an undirected graph, if there is an edge between vertex A and vertex B, you can traverse from A to B and from B to A without restriction.
Undirected graphs can represent symmetrical relationships; for instance, if person A is friends with person B, then person B is also friends with person A.
Common applications of undirected graphs include modeling social networks, transportation systems, and communication networks where the direction of connections is irrelevant.
The complete undirected graph with n vertices has exactly $$\frac{n(n-1)}{2}$$ edges, as every vertex is connected to every other vertex.
Two undirected graphs are said to be isomorphic if there exists a one-to-one mapping between their vertices that preserves the connectivity structure.
Review Questions
How does an undirected graph differ from a directed graph in terms of relationships between vertices?
In an undirected graph, the edges do not have a direction, which means that the relationship between any two vertices is mutual. For example, if vertex A is connected to vertex B, then it implies that B is also connected to A. In contrast, directed graphs have edges with a specific direction, meaning that a connection from A to B does not necessarily imply a connection from B to A. This distinction affects how relationships are represented and analyzed in various applications.
Discuss how the concept of degree applies to undirected graphs and its significance in understanding the structure of such graphs.
The degree of a vertex in an undirected graph is the number of edges incident to it. Understanding the degree helps analyze the connectivity and overall structure of the graph. For instance, vertices with high degrees may indicate central nodes in networks, while those with low degrees might represent peripheral nodes. This concept is significant when studying properties like network robustness, social dynamics, or transportation efficiency since it highlights which nodes play critical roles in connectivity.
Evaluate how undirected graphs can be utilized in real-world applications and the implications of their properties on these systems.
Undirected graphs are widely used in various real-world applications such as social networks, where connections between users (friends) are mutual, and transportation networks, where routes are bidirectional. The properties of undirected graphs influence these systems significantly; for instance, in social networks, high-degree vertices (influential individuals) can affect information flow and trends. In transportation networks, understanding connectivity helps optimize routes and manage traffic flow. The ability to analyze and manipulate these graphs can lead to improved efficiency and insights into the behavior of complex systems.