Cliques in graphs with bounded minimum degree

Let k r ( n , d ) be the minimum number of r-cliques in graphs with n vertices and minimum degree d. We evaluate k r ( n , d ) for n / 2 ⩽ d ⩽ 4 n / 5 and some other cases. Moreover, we give a construction, which we conjecture to give all extremal graph (subject to certain conditions on n, d and r).