Degree

Definition

The degree of a vertex $v$ denoted as $\mathrm{deg}(v)$ 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, $\mathrm{deg}(A)=2$ and $\mathrm{deg}(B)=\mathrm{deg}(C)=1$.

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.

Limitations

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

Definition (Minimum Degree of a Graph)

Let

\delta(G) = \min_{v \in V} \deg(v) $$ denote the minimum degree of any vertex $v \in V$ for [[Graph]] $G$.