Walk and Path  Graph Theory  Discrete Mathematics  Summary and Q&A
TL;DR
A walk is a sequence of vertices and edges in a graph where edges are visited once, while a path is a type of walk where both edges and vertices are not repeated.
Key Insights
 🦔 Walks in graph theory are finite sequences of vertices and edges, with a restriction on the repetition of edges, but not on vertices.
 ❤️🩹 An open walk has different starting and ending vertices, while a closed walk has the same starting and ending vertex.
 🚶 A path is a specific type of walk that is also an open walk, with no repeated edges or vertices.
 🦔 In a path, both edges and vertices must not be visited more than once.
 👻 The distinction between walks and paths allows for more precise analysis of graph structures.
 🌍 Graph theory concepts can be applied to various realworld scenarios, such as network routing or social network analysis.
 🚶 The understanding of walks and paths is fundamental for further exploration of graph theory and its applications.
Questions & Answers
Q: What is a walk in graph theory?
A walk in graph theory refers to a finite sequence of vertices and edges in a graph, where each edge is incident with the vertices preceding and following it. The edges must not be visited more than once, but there is no restriction on the vertices.
Q: What is the difference between an open walk and a closed walk?
An open walk is a type of walk where the starting and ending vertices are different. On the other hand, a closed walk has the same starting and ending vertex.
Q: What is a path in graph theory?
A path is a type of walk in graph theory that is both an open walk and where both edges and vertices are not repeated. It is a sequence of vertices and edges that starts and ends with different vertices, and no vertex or edge is visited more than once.
Q: Can a walk have repeated vertices but nonrepeated edges?
Yes, a walk in graph theory can have repeated vertices as long as the edges are not repeated. The restriction is on the edges, not the vertices.
Summary & Key Takeaways

A walk is a sequence of vertices and edges in a graph where edges are visited once, while vertices can be repeated.

An open walk is a walk where the starting and ending vertices are different, while a closed walk has the same starting and ending vertex.

A path is a type of walk that is also an open walk, where both edges and vertices are not repeated.