Learning to fuse vision and language information and representing them is an important research problem with many applications. Recent progresses have leveraged the ideas of pre-training (from language modeling) and attention layers in Transformers to learn representation from datasets containing images aligned with linguistic expressions that describe the images. In this paper, we propose learning representations from a set of implied, visually grounded expressions between image and text, automatically mined from those datasets. In particular, we use denotation graphs to represent how specific concepts (such as sentences describing images) can be linked to abstract and generic concepts (such as short phrases) that are also visually grounded. This type of generic-to-specific relations can be discovered using linguistic analysis tools. We propose methods to incorporate such relations into learning representation. We show that state-of-the-art multimodal learning models can be further improved by leveraging automatically harvested structural relations. The representations lead to stronger empirical results on downstream tasks of cross-modal image retrieval, referring expression, and compositional attribute-object recognition. Both our codes and the extracted denotation graphs on the Flickr30K and the COCO datasets are publically available on https://sha-lab.github.io/DG.