In this work, we present graph star net (GraphStar), a novel and unified graph neural net architecture which utilizes message-passing relay and attention mechanism for multiple prediction tasks - node classification, graph classification and link prediction. GraphStar addresses many earlier challenges facing graph neural nets and achieves non-local representation without increasing the model depth or bearing heavy computational costs. We also propose a new method to tackle topic-specific sentiment analysis based on node classification and text classification as graph classification. Our work shows that 'star nodes' can learn effective graph-data representation and improve on current methods for the three tasks. Specifically, for graph classification and link prediction, GraphStar outperforms the current state-of-the-art models by 2-5% on several key benchmarks.