Since Transformers can alleviate some critical and fundamental problems of graph neural networks (GNNs), such as over-smoothing, over-squashing and limited expressiveness, they have been successfully applied to graph representation learning and achieved impressive results. However, although there are many works dedicated to make graph Transformers (GTs) aware of the structure and edge information by specifically tailored attention forms or graph-related positional and structural encodings, few works address the problem of how to construct high-performing GTs with modules of GNNs and Transformers. In this paper, we propose a novel graph Transformer with local and global operators interleaving (LGI-GT), in which we further design a new method propagating embeddings of the [CLS] token for global information representation. Additionally, we propose an effective message passing module called edge enhanced local attention (EELA), which makes LGI-GT a full-attention GT. Extensive experiments demonstrate that LGI-GT performs consistently better than previous state-of-the-art GNNs and GTs, while ablation studies show the effectiveness of the proposed LGI scheme and EELA. The source code of LGI-GT is available at https://github.com/shuoyinn/LGI-GT.