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Large Vision-Language Models (LVLMs) have demonstrated remarkable performance across diverse tasks. Despite great success, recent studies show that LVLMs encounter substantial limitations when engaging with visual graphs. To study the reason behind these limitations, we propose VGCure, a comprehensive benchmark covering 22 tasks for examining the fundamental graph understanding and reasoning capacities of LVLMs. Extensive evaluations conducted on 14 LVLMs reveal that LVLMs are weak in basic graph understanding and reasoning tasks, particularly those concerning relational or structurally complex information. Based on this observation, we propose a structure-aware fine-tuning framework to enhance LVLMs with structure learning abilities through three self-supervised learning tasks. Experiments validate the effectiveness of our method in improving LVLMs’ performance on fundamental and downstream graph learning tasks, as well as enhancing their robustness against complex visual graphs.