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  "body": "\n## Statement\n\nLet $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $V$ and [[edge set|0WNGQBPPQX]] $E$.\nThen\n$$\\sum_{v \\in V} \\deg(v) = 2\\, |E|.$$\n\n## Remarks\n\nProof: count the pairs $(v, e)$ with $v \\in e$. Each edge has two ends, and each vertex $v$ lies in $\\deg(v)$ edges. This is [[double counting|0F8YBHF2W3]].\nA direct consequence is that the [[number of odd-degree vertices is even|0QQQQCPZA8]].\nFor a $3$-regular graph on $n$ vertices, it gives $3n = 2 |E|$, so $n$ is even.\n\n## Explanations\n\n### L1 human\n\nAt a party, add up the number of hands each person shakes.\nEach handshake is counted twice, once for each of the two people, so the total is twice the number of handshakes.\n",
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  "name": "Handshaking lemma",
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