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Learning the underlying distribution of molecular graphs and generating high-fidelity samples is a fundamental research problem in drug discovery and material science. However, accurately modeling distribution and rapidly generating novel molecular graphs remain crucial and challenging goals. To accomplish these goals, we propose a novel Conditional Diffusion model based on discrete Graph Structures (CDGS) for molecular graph generation. Specifically, we construct a forward graph diffusion process on both graph structures and inherent features through stochastic differential equations (SDE) and derive discrete graph structures as the condition for reverse generative processes. We present a specialized hybrid graph noise prediction model that extracts the global context and the local node-edge dependency from intermediate graph states. We further utilize ordinary differential equation (ODE) solvers for efficient graph sampling, based on the semi-linear structure of the probability flow ODE. We also combine the solvers with gradient guidance from the molecule property predictor for similarity-constrained molecule optimization. Experiments on diverse datasets validate the effectiveness of our framework. Particularly, the proposed method still generates high-quality molecular graphs in a limited number of steps.