Definition:Walk (Graph Theory)

Definition
A walk on a graph is an alternating series of vertices and edges, beginning and ending with a vertex, in which each edge is incident with the vertex immediately preceding it and the vertex immediately following it.

A walk between two vertices $$u$$ and $$v$$ is called a $$u$$-$$v$$ walk.

A walk is closed if the first vertex is the same as the last. Otherwise it is open.

Length
The length of a walk is the number of edges it has, counting repeated edges as many times as they appear.

Trails and Paths
A trail is a walk in which all edges are distinct.

A path is a walk in which all vertices are distinct.

Circuits and Cycles
A closed trail is called a circuit.

A closed path is called a cycle.

Differences in Terminology
Some sources refer to a walk as a path, and use the term simple path to define what we have here as a path.