What defines a walk in graph theory?

In graph theory, a walk is defined as a sequence of vertices and edges where you can move from one vertex to another along the edges of the graph, and you are allowed to visit the same vertex and edge more than once. This means that a walk can include repeated vertices and repeated edges, allowing for maximum flexibility in traversal within the graph.

Understanding this definition highlights why the option describing a route that includes repeated edges and vertices accurately represents what a walk is. It's important to differentiate a walk from other terms such as a trail, which is a route that does not repeat edges, or a path, which does not repeat vertices.

The other options describe different concepts in graph theory rather than what constitutes a walk. For instance, a route with no repeated vertices would be more aligned with the definition of a path, while a route using all edges exactly once describes an Eulerian path or circuit, which has its own specific conditions. Recognizing the nuances in these definitions is crucial for accurately identifying concepts in graph theory.