a simple graph has 7 vertices and 10 edges. each vertex has degree at most 3. Find the number of vertices of degree 2

Question

I have created two equations based on this statement, let x be the number of vertices with degree 1, y be number of vertices with degree 2, z be number of vertices with degree 3.

$x+2y+3z=20$ from the handshake theorem

and $x+y+z=7$

But am not sure where to proceed. Can anyone help me out. Thanks.

Answer 1

We have $$ x+2y+3z = 20 \tag1 $$$$ x+y+z = 7 \tag2 . $$ Multiplying $(2)$ by $3$ and subtracting $(1)$ from it we get $$ 2x + y = 1. $$ Since $x,y$ are integers we conclude that $x=0$, $y=1$ and so $z = 6$.