how to Compute the number of pairwise non-isomorphic 7-regular graphs on 10 vertices?

Question

Compute the number of pairwise non-isomorphic 7-regular graphs on 10 vertices?

Answer 1

You do that by looking at the complement, the graph of missing edges.

Answer 2

If you're interested in this question from a computational mathematics point of view (and this is a specific instance of a broader problem), then this can be computed using Brendan McKay's geng (part of the gtools package, which can be downloaded with nauty). Specifically:

geng -n 10 -d7 -D7

computes the number of non-isomorphic graphs with $10$ vertices, minimum degree $7$ and maximum degree $7$.

If you want to understand why the number is what it is, Hendrik Jan's hint is excellent.