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Math Help - Graph Theory

  1. #1
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    Graph Theory

    Let G be a r-regular graph on n vertices with e edges. Prove one of n or r must divide e.

    The only thing that I can think of to apply to this problem is that a r-regular graph has every vertex at r degrees.



    Thanks in advance!!
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  2. #2
    Junior Member guildmage's Avatar
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    Maybe you can use the handshaking lemma for this. Recall that \sum\limits_{v \in V} {d(v)} = 2e, where d(v) is the degree of a vertex v. Since the graph is r-regular and has n vertices, then we have nr=2e. But if n and r do not divide e (at the same time). Then nr does not divide e. This is a contradiction. Thus either n or r must divide e.
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