Final answer:
In any digraph, the sum of the in-degrees of all vertices is equal to the sum of their out-degrees, and this sum is equal to the number of edges in the digraph.
Step-by-step explanation:
In any digraph (directed graph), the sum of the in-degrees of all vertices is equal to the sum of their out-degrees, and this sum is equal to the number of edges in the digraph.
To prove this, we can use Kirchhoff's first rule, which states that the sum of all currents entering a junction must equal the sum of all currents leaving the junction. In the context of a digraph, the junctions represent the vertices.
Since the in-degree of a vertex represents the number of edges entering the vertex, and the out-degree represents the number of edges leaving the vertex, we can apply Kirchhoff's first rule to prove that the sum of the in-degrees is equal to the sum of the out-degrees and the number of edges.