# Definition:Regular Graph

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## Definition

Let $G = \struct {V, E}$ be an simple graph whose vertices all have the same degree $r$.

Then $G$ is called **regular of degree $r$**, or **$r$-regular**.

## Examples

### Incomplete $0$-Regular

The $0$-regular graphs which are not complete are the edgeless graphs $N_n$ of order $n$ for $n > 1$.

For example, $N_2$:

### Incomplete $1$-Regular

An example of a $1$-regular graph which is not complete is shown below:

### Incomplete $2$-Regular

The $2$-regular graphs which are not complete are the cycle graphs $C_n$ of order $n$ for $n > 3$.

For example, $C_4$:

Note that $C_3$ is both $2$-regular and complete.

### Incomplete $3$-Regular

An example of a $3$-regular graph which is not complete is shown below:

## Also see

- Definition:Edgeless Graph: a
**$0$-regular graph**. - Definition:Cycle Graph: a
**$2$-regular graph**. - Definition:Cubic Graph: a
**$3$-regular graph**.

- Results about
**regular graphs**can be found here.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Chapter $2$: Elementary Concepts of Graph Theory: $\S 2.1$: The Degree of a Vertex - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**regular graph**