As　a　kind　of　generalization　of　Transformers　in　the　graph　domain,　Global　Graph　Transformers　are　good　at　learning　distant　knowledge　by　directly　doing　information　interactions　on　complete　graphs,　which　differs　from　Local　Graph　Transformers　interacting　on　the　original　structures.　However,　we　find　that　most　prior　works　focus　only　on　graph-level　tasks　(e.g.,　graph　classification)　and　few　Graph　Transformer　models　can　effectively　solve　node-level　tasks,　especially　semi-supervised　node　classification,　which　obviously　has　important　practical　significance　due　to　the　limitation　and　expensiveness　of　these　node　labels.　In　order　to　fill　this　gap,　this　paper　first　summarizes　the　theoretical　advantages　of　Graph　Transformers.　And　based　on　some　exploring　experiments,　we　give　some　discussions　on　the　main　cause　of　their　poor　practical　performance　in　semi-supervised　node　classifications.　Secondly,　based　on　this　analysis,　we　design　a　three-stage　homogeneity　augmentation　framework　and　propose　a　Semi-Global　Graph　Transformer.　Considering　both　global　and　local　perspectives,　the　proposed　model　combines　various　technologies　including　self-distillation,　pseudo-label　filtering,　pre-training　and　fine-tuning,　and　metric　learning.　Furthermore,　it　simultaneously　enhances　the　structure　and　the　optimization,　improving　its　effectiveness,　scalability,　and　generalizability.　Finally,　extensive　experiments　on　seven　public　homogeneous　and　heterophilous　graph　benchmarks　show　that　the　proposed　method　can　achieve　competitive　or　much　better　results　compared　to　many　baseline　models　including　state-of-the-arts.　©　2023　World　Scientific　Publishing　Company.