Matching between questions and suitable users is an appealing and challenging problem in the research area of community question answering (CQA). Usually, different from the traditional recommendation systems where a user has only a single role, each user in CQA can play two different roles (dual roles) simultaneously: as a requester and as an answerer. For different roles, users usually have varying interests and expertise in different topics and knowledge domains, which is rarely addressed in the previous methods. Besides, based on an explicit single link between two users, existing methods cannot capture implicit associations between their possibly similar roles. Therefore, in this paper, we propose the structure of a dual role graph and employ the link prediction approach to make CQA recommendation on the graph. Moreover, we develop a Dual Role Neural Graph auto-encoder (DRNGae) framework, which can: 1) encode the dual role graph structure to capture the implicit dual role correlation by propagating high-order information embeddings of graph neural network; 2) learn variable weights with the dual role feature preferences from dual role content information by self-attention mechanism; 3) reconstruct the graph structure to predict the possible interaction links. Experimental studies on real-world datasets verify our design and prove that our model achieves significantly better performance than baselines in link prediction (95.3% AUC, 96.2% AP on Citeseer dataset) and CQA recommendation (79.5% recall@25, 76.7% ndcg@25 on Yahoo! answer dataset).