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Chapter 14: Problem 59

What are adjacent vertices? If two vertices are near each other in a graph, are they necessarily adjacent? Explain your answer.

Short Answer

Expert verified
Adjacent vertices in a graph are vertices connected by the same edge. They are called 'adjacent' not because they are 'near' each other but because there's a direct edge connecting them. Hence, two vertices being near each other in a graph does not necessarily mean they are adjacent.

Step by step solution

01

Definition of Adjacent Vertices

In the domain of graph theory, two vertices are said to be adjacent if they are both endpoints of a same edge. Simply put, if there is a direct edge connecting vertex u and vertex v, these vertices are adjacent.
02

Explanation of Nearness Vs Adjacency

The concept of 'nearness' is subjective and not clearly defined in the context of graph theory. In a graph, vertices are considered 'near' based on the path length or number of edges between them, whereas they are 'adjacent' based on being endpoints of the same edge. Therefore, being physically 'near' does not imply adjacency. For two vertices to be adjacent, they must be connected by an edge.
03

Conclusion

So, not all vertices close to each other in a graph are necessarily adjacent. They are adjacent only if there is a direct edge connecting them.

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