# 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**.

To describe a **walk** on a simple graph it is sufficient to list just the vertices in order, as the edges (being unique between vertices) are unambiguous.

### Closed

A **closed walk** is a walk whose first vertex is the same as the last.

That is, it is a walk which ends where it starts.

### Open

An **open walk** is a walk whose first vertex and last vertex are distinct.

That is, it is a walk which ends on a different vertex from the one where it starts.

## Length

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

A walk is said to be of **infinite length** if and only if it has infinitely many edges.

## Also known as

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

## Also see

- Definition:Trail: a
**walk**in which all edges are distinct.

- Definition:Path (Graph Theory): a
**walk**in which all vertices are distinct.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): $\S 2.3$: Connected Graphs - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**walk** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**walk** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**walk** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**walk**(in graph theory)