Internal and External Degrees of a Vertex

Topics: Vertex Degree - Graph Theory

---

(definition)

When we have a vertex $v_i$ connected to an arc, we define:

• Its internal degree, denoted $d^{-}(v_i)$, as the amount of arcs that point to said vertex.
• Its external degree, denoted $d^{+}(v_i)$, as the amount of arcs that point away from said vertex.

Sum Theorem §

Given a digraph with $n$ vertices, the sum of the internal degrees of all its vertices is equal to the sum of all the external degrees of all its vertices:

$$\sum _{i=1}^n{d}^{+}(v_i)=\sum _{i=1}^n{d}^{-}(v_i)=e$$

Both of these sums is equal to the amount of edges of the digraph.