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Proof

(related to Lemma: Size of an \(r\)-regular Graph with \(n\) Vertices)

By hypothesis, \(G(V,E)\) is an \(r\)-regular graph with \(n\) vertices.
Therefore, the total sum of all degrees of the vertices is \(nr\).
By the handshaking lemma for finite graphs, the number of edges (the size of \(G\)) is half the number of the total sum, i.e. \(|E|={nr}/2.\)
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References

Bibliography

Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000