Graph Transformer has recently received wide attention in the research community with its outstanding performance, yet its structural expressive power has not been well analyzed. Inspired by the connections between Weisfeiler-Lehman (WL) graph isomorphism test and graph neural network (GNN), we introduce \textbf{SEG-WL test} (\textbf{S}tructural \textbf{E}ncoding enhanced \textbf{G}lobal \textbf{W}eisfeiler-\textbf{L}ehman test), a generalized graph isomorphism test algorithm as a powerful theoretical tool for exploring the structural discriminative power of graph Transformers. We theoretically prove that the SEG-WL test is an expressivity upper bound on a wide range of graph Transformers, and the representational power of SEG-WL test can be approximated by a simple Transformer network arbitrarily under certain conditions. With the SEG-WL test, we show how graph Transformers' expressive power is determined by the design of structural encodings, and present conditions that make the expressivity of graph Transformers beyond WL test and GNNs. Moreover, motivated by the popular shortest path distance encoding, we follow the theory-oriented principles and develop a provably stronger structural encoding method, Shortest Path Induced Subgraph (\textit{SPIS}) encoding. Our theoretical findings provide a novel and practical paradigm for investigating the expressive power of graph Transformers, and extensive synthetic and real-world experiments empirically verify the strengths of our proposed methods.

翻译：图Transformer以其卓越的性能近期在研究社区中获得广泛关注，但其结构表达能力尚未得到充分分析。受Weisfeiler-Lehman（WL）图同构测试与图神经网络（GNN）之间联系的启发，我们引入**SEG-WL测试**（**结构编码增强的全局Weisfeiler-Lehman测试**），这是一种广义图同构测试算法，可作为探索图Transformer结构判别能力的强大理论工具。我们从理论上证明：SEG-WL测试是众多图Transformer的表达能力上界，并且在一定条件下，SEG-WL测试的表征能力可由一个简单Transformer网络任意逼近。借助SEG-WL测试，我们揭示了图Transformer的表达能力如何由结构编码的设计决定，并给出了使图Transformer表达能力超越WL测试和GNN的条件。此外，受流行的最短路径距离编码启发，我们遵循理论导向原则，开发了一种可证明更强的结构编码方法——最短路径诱导子图（SPIS）编码。我们的理论发现为探究图Transformer的表达能力提供了新颖而实用的范式，大量合成实验与真实实验也实证验证了所提方法的优势。