Proof
(related to Lemma: Handshaking Lemma for Finite Graphs)
 In any graph \(G(V,E,\gamma)\), each edge has two ends.
 Thus, it contributes exactly \(2\) to the sum of the vertex degrees, which is welldefined, if the graph is finite.
 It follows $$\sum_{v\in V}\deg_G(v)=2E.$$
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References
Bibliography
 Aldous Joan M., Wilson Robin J.: "Graphs and Applications  An Introductory Approach", Springer, 2000