# 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

• Results about regular graphs can be found here.