In this paper, we study the Sturm--Liouville differential operators with a constant delay and transmission(discontinuity,jump) boundary conditions. We investigate the properties of the asymptotic form of eigenvalues and characteristic functions of the operators. An inverse spectral problem of recovering these type operators from the spectra of two boundary value problems, first with Dirichlet-Dirichlet and second with Dirichlet-Neumann boundary conditions, is studied. We will construct the integral equations of the potential, using this integral equations we will proved uniqueness of the potential. Indirectly, we will reconstruct potential from two spectra.
