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I'm doing a question in my text that asks me to find how many regions a graph has if it has 6 vertices all of which has degree 4. I know i'm supposed to use the euler theorem of $r=e-v+2$, but to use this I need to find the number of edges first. So how do I find the number of edges in a graph if I know how many vertices there are and their degrees?

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1 Answer

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The sum of the vertex degree values is twice the number of edges, because each of the edges has been counted from both ends.

In your case $6$ vertices of degree $4$ mean there are $(6\times 4) / 2 = 12$ edges.

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