Vertices $1, 3, 5,7,9$ have degree $8$ and vertices $2, 4, 6, 8$ have degree $7$.
We know tht in a graph Sum of degrees $= 2\times \text{No. of edges}$
Let $X =$ degree of vertex $10$
$8 + 7 + 8 + 7 + 8 + 7 + 8 + 7 + 8 + X = 2\times 37$
$68 + X =74$
$X=6$
Hence,Degree of vertex $10$ is $6$.
Hence, Option (B) $6$ is the correct choice.