b. An undirected simple graph in a star topology with one central vertex v such that E = {(v, x)/x e V, v#x}. See figure below for a "star" graph with six vertices. E Reflexive Symmetric Antisymmetric Transitive
b. An undirected simple graph in a star topology with one central vertex v such that E = {(v, x)/x e V, v#x}. See figure below for a "star" graph with six vertices. E Reflexive Symmetric Antisymmetric Transitive
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please explain in detail
![### Graph Theory Concepts: Star Topology
#### Explanation
**Graph Description (Part b):**
An undirected simple graph in a star topology with one central vertex \( v \) is described as follows:
\[ E = \{(v, x) | x \in V, v \neq x\} \]
This notation describes the set of edges \( E \) in the graph, where each edge connects the central vertex \( v \) to another vertex \( x \) in the set of vertices \( V \), such that \( v \neq x \).
**Graph Visualization:**
Below is a diagram of a "star" graph with six vertices. This graph displays one central vertex \( v \) connected to five other vertices.
#### Properties
- **Reflexive**
- **Symmetric**
- **Antisymmetric**
- **Transitive**
#### Diagram Explanation
The diagram illustrates a star topology with six vertices:
- The central node, labeled \( v \), is connected to five peripheral nodes.
- Each line represents an edge connecting the central node \( v \) to one of the peripheral nodes.
The graph has the following properties:
- **Reflexive:** This property does not apply to this graph as there are no self-loops.
- **Symmetric:** The graph is symmetric because if there is an edge between \( v \) and any vertex \( x \), there is also an edge between \( x \) and \( v \).
- **Antisymmetric:** This property does not typically apply to undirected graphs as antisymmetry refers to directed graphs where if \( (a, b) \) is in the relation and \( (b, a) \) is in the relation, then \( a = b \).
- **Transitive:** This property does not apply here as transitivity typically applies to directed graphs.
This star topology arrangement is often used in network designs where a central node relays information to and from other nodes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F416064f9-c57b-46ee-9b50-14b257515654%2F078bc8e2-9367-4a9a-9510-8205060c7fda%2F4uc89kc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graph Theory Concepts: Star Topology
#### Explanation
**Graph Description (Part b):**
An undirected simple graph in a star topology with one central vertex \( v \) is described as follows:
\[ E = \{(v, x) | x \in V, v \neq x\} \]
This notation describes the set of edges \( E \) in the graph, where each edge connects the central vertex \( v \) to another vertex \( x \) in the set of vertices \( V \), such that \( v \neq x \).
**Graph Visualization:**
Below is a diagram of a "star" graph with six vertices. This graph displays one central vertex \( v \) connected to five other vertices.
#### Properties
- **Reflexive**
- **Symmetric**
- **Antisymmetric**
- **Transitive**
#### Diagram Explanation
The diagram illustrates a star topology with six vertices:
- The central node, labeled \( v \), is connected to five peripheral nodes.
- Each line represents an edge connecting the central node \( v \) to one of the peripheral nodes.
The graph has the following properties:
- **Reflexive:** This property does not apply to this graph as there are no self-loops.
- **Symmetric:** The graph is symmetric because if there is an edge between \( v \) and any vertex \( x \), there is also an edge between \( x \) and \( v \).
- **Antisymmetric:** This property does not typically apply to undirected graphs as antisymmetry refers to directed graphs where if \( (a, b) \) is in the relation and \( (b, a) \) is in the relation, then \( a = b \).
- **Transitive:** This property does not apply here as transitivity typically applies to directed graphs.
This star topology arrangement is often used in network designs where a central node relays information to and from other nodes.
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