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Let D be a finite digraph. If id(x) >0 for every x element of V(D), then D contains a cycle.

Let D be a finite digraph. If id(x) >0 for every x element of V(D), then D contains a cycle.

Advanced Engineering Mathematics
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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**Understanding the Relationship Between In-Degree and Cycles in Directed Graphs** Consider a finite directed graph (digraph) \(D\). If the in-degree \(id(x)\) of each vertex \(x\) in the set of vertices \(V(D)\) is greater than 0, then it implies that every vertex in \(D\) has at least one incoming edge. Under this condition, we can conclude that the digraph \(D\) contains at least one cycle. **Key Concepts:** 1. **Directed Graph (Digraph)**: - A graph where the edges have a direction, represented by arrows pointing from one vertex to another. 2. **In-Degree (id(x))**: - The number of edges directed into a vertex \(x\). 3. **Cycle**: - A path in which the starting and ending vertices are the same, and no vertices are repeated along the path except for the start/end vertex. **Implication**: - If every vertex \(x\) in \(D\) has \(id(x) > 0\), there are no vertices which are isolated from incoming connections. This condition ensures that there is a cycle within the digraph, meaning there is a closed loop path that can be formed within \(D\). Understanding these principles is important for analyzing the structure and properties of directed graphs, especially in fields such as graph theory, computer science, and network analysis.
Transcribed Image Text:**Understanding the Relationship Between In-Degree and Cycles in Directed Graphs** Consider a finite directed graph (digraph) \(D\). If the in-degree \(id(x)\) of each vertex \(x\) in the set of vertices \(V(D)\) is greater than 0, then it implies that every vertex in \(D\) has at least one incoming edge. Under this condition, we can conclude that the digraph \(D\) contains at least one cycle. **Key Concepts:** 1. **Directed Graph (Digraph)**: - A graph where the edges have a direction, represented by arrows pointing from one vertex to another. 2. **In-Degree (id(x))**: - The number of edges directed into a vertex \(x\). 3. **Cycle**: - A path in which the starting and ending vertices are the same, and no vertices are repeated along the path except for the start/end vertex. **Implication**: - If every vertex \(x\) in \(D\) has \(id(x) > 0\), there are no vertices which are isolated from incoming connections. This condition ensures that there is a cycle within the digraph, meaning there is a closed loop path that can be formed within \(D\). Understanding these principles is important for analyzing the structure and properties of directed graphs, especially in fields such as graph theory, computer science, and network analysis.
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