Generating drug-like molecules that specifically bind to target proteins remains a resource-intensive challenge. Many studies focus on designing effective networks to accurately extract relevant features from target proteins, which can be challenging. Additionally, most target-specific molecule generation methods based on diffusion models process the 3D information of molecules and proteins, necessitating the maintenance of equivariance at each step. This paper proposes TarMGDif, a novel target-specific molecular graph generation model based on a discrete denoising diffusion framework which could handle graph structure. TarMGDif incorporates a global features embedding network that captures ring features to generate chemically valid rings, while the time step of the diffusion model is also learned through this network. Besides, a novel node-to-edge attention module is proposed to capture dependencies between nodes and edges.Extensive experiments conducted on three datasets demonstrate the advanced performance of TarMGDif. Furthermore, through transfer learning, the model generates molecules specifically targeting the DRD2 protein, with the newly designed molecules exhibiting pharmacological properties similar to known inhibitors. These findings underscore the potential of TarMGDif in facilitating the efficient design of target-specific drug-like molecules.