Star graph

Definition

A star graph is a specific type of graph that consists of one central vertex connected directly to several outer vertices, forming a star-like shape. This structure highlights the concept of connectivity and the relationships between vertices, as the central vertex serves as a hub while all other vertices are leaves with only one edge connecting them to the center. The degree of the central vertex is equal to the number of outer vertices, while all other vertices have a degree of one.

5 Must Know Facts For Your Next Test

1. In a star graph with 'n' outer vertices, the central vertex will have a degree of 'n', while each outer vertex will have a degree of 1.
2. Star graphs are a type of tree graph, specifically a tree with one internal node and multiple leaves.
3. Star graphs are commonly used in network topology to represent hubs and spoke models, where the hub connects multiple nodes.
4. The star graph is denoted as $S_n$, where 'n' represents the number of outer vertices connected to the central vertex.
5. Star graphs are often utilized in algorithms for their efficient communication and connection properties due to their centralized structure.

Review Questions

• How does the structure of a star graph influence its properties in terms of connectivity and vertex degree?
  • The structure of a star graph significantly influences its properties by centralizing connectivity around one central vertex. This vertex has a degree equal to the number of outer vertices, highlighting its role as a hub. The outer vertices have only one connection each, which means they rely entirely on the central vertex for communication. This creates a highly connected graph in terms of the central node while maintaining simplicity and limited connections among the outer nodes.
• Discuss how star graphs can be applied in real-world scenarios, particularly in network design.
  • Star graphs are highly relevant in network design, especially in scenarios like computer networks where a central router connects multiple devices. This hub-and-spoke model allows for efficient data management and communication since all traffic passes through the central point. Furthermore, if one connection fails, it does not disrupt the entire network, as only that specific device loses connectivity while others remain unaffected. This reliability makes star graphs popular in designing local area networks (LANs).
• Evaluate the advantages and disadvantages of using star graphs compared to complete graphs in terms of efficiency and complexity.
  • Using star graphs offers significant advantages over complete graphs, primarily due to their simplicity and efficiency. Star graphs require fewer edges than complete graphs, making them less complex and easier to manage, especially as the number of vertices increases. However, this simplicity comes at the cost of vulnerability; if the central vertex fails, all outer vertices lose connectivity. In contrast, complete graphs provide maximum connectivity at the expense of higher complexity and resource consumption due to their many edges. Therefore, the choice between these two types depends on specific needs for connectivity versus efficiency.

Related terms

Vertex: A fundamental unit in a graph representing a point where edges meet.

Degree:

The number of edges connected to a vertex, indicating its connectivity within the graph.

Complete Graph:

A type of graph in which every pair of distinct vertices is connected by a unique edge.