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Lemma: Handshaking Lemma for Finite Digraphs
In any finite digraph \(D=(V,E,\alpha,\omega)\), the sum of all outer degrees and the sum of all inner degrees are both equal to the number of edges:
\[\sum_{v\in V}d_D^+(v)=\sum_{v\in V}d_D^(v)=E.\]
In particular, the sum of all degrees is even:
\[\sum_{v\in V}d_D(v)=2E.\]
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References
Bibliography
 Krumke S. O., Noltemeier H.: "Graphentheoretische Konzepte und Algorithmen", Teubner, 2005, 1st Edition