Image-text matching aims to measure the similarities between images and textual descriptions, which has made great progress recently. The key to this cross-modal matching task is to build the latent semantic alignment between visual objects and words. Due to the widespread variations of sentence structures, it is very difficult to learn the latent semantic alignment using only global cross-modal features. Many previous methods attempt to learn the aligned image-text representations by the attention mechanism but generally ignore the relationships within textual descriptions which determine whether the words belong to the same visual object. In this paper, we propose a graph attentive relational network (GARN) to learn the aligned image-text representations by modeling the relationships between noun phrases in a text for the identity-aware image-text matching. In the GARN, we first decompose images and texts into regions and noun phrases, respectively. Then a skip graph neural network (skip-GNN) is proposed to learn effective textual representations which are a mixture of textual features and relational features. Finally, a graph attention network is further proposed to obtain the probabilities that the noun phrases belong to the image regions by modeling the relationships between noun phrases. We perform extensive experiments on the CUHK Person Description dataset (CUHK-PEDES), Caltech-UCSD Birds dataset (CUB), Oxford-102 Flowers dataset and Flickr30K dataset to verify the effectiveness of each component in our model. Experimental results show that our approach achieves the state-of-the-art results on these four benchmark datasets.