  We develop techniques to construct a series of sparse polyhedral
approximations of the semidefinite cone. Motivated by the semidefinite (SD)
bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of
SD bases so as to keep the sparsity of the matrices composing it. We prove that
the polyhedral approximation using our expanded SD bases contains the set of
all diagonally dominant matrices and is contained in the set of all scaled
diagonally dominant matrices. We also prove that the set of all scaled
diagonally dominant matrices can be expressed using an infinite number of
expanded SD bases. We use our approximations as the initial approximation in
cutting plane methods for solving a semidefinite relaxation of the maximum
stable set problem. It is found that the proposed methods with expanded SD
bases are significantly more efficient than methods using other existing
approximations or solving semidefinite relaxation problems directly.
