Multi-Trait Essay Scoring (MTES) is crucial for educational systems, yet existing MTES models ignore genre-specific writing differences and fail to capture explicit structures such as argument hierarchies and story-causal chains. To address this gap, this paper proposes a novel framework GeGES, which uses few-shot LLM prompting to extract an Argumentation Structure Graph, a Response-Evidence Graph, and a StoryGrammar Causal Graph for argumentative, source-dependent, and narrative essays, respectively; then encodes these graphs with a Graph Attention Network enhanced by multi-view contrastive learning; and finally feeds the resulting structural tokens, together with the original text, into a fine-tuned LLM to autoregressively generate trait scores. On the ASAP++ benchmark, GeGES achieves an average QWK of 0.725, surpassing the previous best model, and yields notable gains on structure-sensitive traits such as Content and Organization, confirming the effectiveness of jointly modeling genre-wise distinctions and structural information.