The zero-dilation index d(A) of a square matrix A is defined as the maximum size k of a zero matrix whichcan be dilated to A. The purpose of this project is to find the upper bound of d(A) and to characterize thoseA's which attain this bound among two classes of matrices, namely, the Sn-matrices and companion matrices.In this project, we conjecture that if A is an Sn-matrix or an n-by-n companion matrix, then d(A) is at mostthe smallest integer greater than or equal to n/2. We also want to characterize those A's for which the upperbound is attained. Among other things, we conjecture that, for an odd n, the Sn-matrix A is such thatd(A)=(n+1)/2 if and only if it is unitarily similar to -A, and, for an even n, every n-by-n companion matrix Ahas d(A) equal to n/2.