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A graph with 21 edges has seven vertices of degree 1, three of degree 2, seven of degree 3 and the rest of degree 4. How many vertices does it have?

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3 Answers 3

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The sum of the degrees is twice the number of edges, so just solve for $x$, $$7\cdot 1 + 3 \cdot 2 + 7 \cdot 3 + x \cdot 4 = 2 \cdot 21$$

Then $7 + 3 + 7 + x$ is the number of vertices.

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Suppose that it has $n$ vertices of degree $4$; then the sum of the degrees is $$7+3\cdot2+7\cdot3+4n=4n+34\;.$$ On the other hand, you know that the sum of the degrees of the vertices is twice the number of edges, so ... ?

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Sum of degrees is equal to 2times the number of edges. So we are given the edges as 21 therefore 21*2=42. So we just solve for y 7*1+3*2+7*3+y*4 = 42 Solving for y= 4x= 42-34 Y=2 Therefore 2 is the number of vertices of degree 4

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