Definition:Walk (Graph Theory)

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

Trail 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 circuit in which no vertex except the first appears more than once is a cycle. An n-cycle is a cycle with $$n$$ vertices.