In dialogue generation, the naturalness of responses is crucial for effective human-machine interaction. Personalized response generation poses even greater challenges, as the responses must remain coherent and consistent with the user's personal traits or persona descriptions. We propose MUDI ($\textbf{Mu}$ltiple $\textbf{Di}$scourse Relations Graph Learning) for personalized dialogue generation. We utilize a Large Language Model to assist in annotating discourse relations and to transform dialogue data into structured dialogue graphs. Our graph encoder, the proposed DialogueGAT model, then captures implicit discourse relations within this structure, along with persona descriptions. During the personalized response generation phase, novel coherence-aware attention strategies are implemented to enhance the decoder's consideration of discourse relations. Our experiments demonstrate significant improvements in the quality of personalized responses, thus resembling human-like dialogue exchanges.