Before revision:  For n-vertex graphs with constant diameter D=O(1), Das-Sarma et al. (STOC 2011, SIAM J. Comput. 2012 ) presented an (implicit) shortcuts lower bound with (As usual, and hide poly-logarithmic factors.) c+d=\Omega(n^{(D-2)/(2D-2)}).
After revision:  For n-vertex graphs with constant diameter D=O(1), Elkin (STOC 2004 ) presented an (implicit) shortcuts lower bound with (As usual, and hide poly-logarithmic factors.) c+d=\Omega(n^{(D-2)/(2D-2)}).