Despite the remarkable accomplishments of graph neural networks (GNNs), they typically rely on task-specific labels, posing potential challenges in terms of their acquisition. Existing work have been made to address this issue through the lens of unsupervised domain adaptation, wherein labeled source graphs are utilized to enhance the learning process for target data. However, the simultaneous exploration of graph topology and reduction of domain disparities remains a substantial hurdle. In this paper, we introduce the Dual Adversarial Graph Representation Learning (DAGRL), which explore the graph topology from dual branches and mitigate domain discrepancies via dual adversarial learning. Our method encompasses a dual-pronged structure, consisting of a graph convolutional network branch and a graph kernel branch, which enables us to capture graph semantics from both implicit and explicit perspectives. Moreover, our approach incorporates adaptive perturbations into the dual branches, which align the source and target distribution to address domain discrepancies. Extensive experiments on a wild range graph classification datasets demonstrate the effectiveness of our proposed method.