Title: Graph Domain Adaptation via Theory-Grounded Spectral Regularization. Abstract: Transfer learning on graphs drawn from varied distributions (domains) is in great demand across many applications. Emerging methods attempt to learn domain-invariant representations using graph neural networks (GNNs), yet the empirical performances vary and the theoretical foundation is limited. This paper aims at designing theory-grounded algorithms for graph domain adaptation (GDA). (i) As the first attempt, we derive a model-based GDA bound closely related to two GNN spectral properties: spectral smoothness (SS) and maximum frequency response (MFR). This is achieved by cross-pollinating between the OT-based (optimal transport) DA and graph filter theories. (ii) Inspired by the theoretical results, we propose algorithms regularizing spectral properties of SS and MFR to improve GNN transferability. We further extend the GDA theory into the more challenging scenario of conditional shift, where spectral regularization still applies. (iii) More importantly, our analyses of the theory reveal which regularization would improve performance of what transfer learning scenario, (iv) with numerical agreement with extensive real-world experiments: SS and MFR regularizations bring more benefits to the scenarios of node transfer and link transfer, respectively. In a nutshell, our study paves the way toward explicitly constructing and training GNNs that can capture more transferable representations across graph domains. Codes are released at https://github.com/Shen-Lab/GDA-SpecReg.