A noncomplete graph $H$ of order $5$ or more with the property that $deg (u) ≠ deg (v)$ for every pair $u, v$ of nonadjacent vertices of $H$. I'm working in the following graph theory excercise:

Give an example or explain why no such example exists of a noncomplete graph $H$ of order $5$ or more with the property that $deg (u) ≠ deg (v)$ for every
  pair $u, v$ of nonadjacent vertices of $H$.

I've reached the conclusion that is impossible that this graph exists because it can not be generated edges infinitely, is there a formalization that could help me to show it? Thanks in advance for any hint or help.

Does
$$
H = ([5], \{(1,2),(2,3),(2,4),(3,4),(3,5),(4,5)\}).
$$
satisfy your requirements?
I found it by considering the complement, which has the property that every pair of adjacent vertices has distinct degrees.
