In recent years, online question-answering (Q&A) platforms, such as Stack Exchange (SE), have become increasingly popular as a source of information and knowledge sharing. Despite the vast amount of information available on these platforms, many questions remain unresolved. In this work, we aim to address this issue by proposing a novel approach to identify unresolved questions in SE Q&A communities. Our approach utilises the graph structure of communication formed around a question by users to model the communication network surrounding it. We employ a property graph model and graph neural networks (GNNs), which can effectively capture both the structure of communication and the content of messages exchanged among users. By leveraging the power of graph representation and GNNs, our approach can effectively identify unresolved questions in SE communities. Experimental results on the complete historical data from three distinct Q&A communities demonstrate the superiority of our proposed approach over baseline methods that only consider the content of questions. Finally, our work represents a first but important step towards better understanding the factors that can affect questions becoming and remaining unresolved in SE communities.