Definition

The degree of a vertex $v$ denoted as $de g (v)$ is the number of edges that $v$ is incident on.

graph LR;
	A((A)) o--o B((B));
	A((A)) o--o C((C));

Here, $de g (A) = 2$ and $de g (B) = de g (C) = 1$ .

Definition: $d -$ regular

A Graph $G$ is $d -$ regular if all vertices have degree $d$ . We can also say that that $G$ is just regular.

For example:

flowchart
	A((A))
	B((B))
	C((C))

is $0 -$ regular,

graph LR; 
	A((A)) x--x B((B));
	B((B)) x--x C((C));
	C((C)) x--x D((D));
	D((D)) x--x A((A));

is $2 -$ regular,

graph 
	A((A)) x--x B((B));
	C((C)) x--x D((D));
	E((E)) x--x F((F));

is $1 -$ regular.

Some graphs you cannot draw. For example, there is no $3 -$ regular graph with $5$ vertices.

Let

δ (G) = v \in V min de g (v)

denote the minimum degree of any vertex $v \in V$ for Graph $G$ .

Let

Δ (G) = v \in V max de g (v)

denote the maximum degree of any vertex in graph $G$ .

For $Δ (G) \in {0, 1}$ , then $∣ E ∣ = 0$ or any connected component is at most Complete Graph $K_{2}$ .
For $Δ (G) = 2$ , then we must have a Path graph or a cycle.