I am looking for NP complete results for cliques in regular graphs. For example is the general problem of determining if a regular graph on n vertices has an n/2 clique NPcomplete? (obviously the question is interesting only if the degree is at least n/2). You could also ask the same question for say regular graphs of degree 3n/4.

$\begingroup$ There is lots of information (though not an answer to your exact question) at en.wikipedia.org/wiki/Clique_problem. $\endgroup$– Hugh ThomasJun 24 '14 at 4:06
Let $G = (V,E)$ be a graph and $c ≤ V $ a positive integer. The independent sets of $G$ are precisely the cliques of the complementary graph $\overline{G}$.
INDEPENDENT SET
INSTANCE: Graph $G=(V,E)$, positive integer $c$.
QUESTION: Does $G$ contain an independent set of size $c$ or more,
The independent set problem remains NPcomplete when restricted to 3regular planar graphs.
Reference:
COMPUTERS AND Intractability, A Guide to the Theory of NPCompleteness
Michael R. Garey / David S. Johnson
page 194195

1$\begingroup$ This answers the question for cliques of specified size, but it doesn't answer it for cliques of size n/2. 3regular graphs have an independent set of size $n/2$ iff they are bipartite. $\endgroup$ Jun 25 '14 at 19:42

$\begingroup$ The key is that the complement of a regular graph is regular. $\endgroup$ Jun 28 at 13:34