Nowadays, the efficient representation of a given shape plays a significant role in pattern recognition field. In this paper, we introduce a novel method to represent 2-D shape as a relation graph which using affine-invariant Fourier. This presentation method has invariance properties, such as scaling, shifting, rotation and starting point. Then edit distance is used to measure the distance between graphs. Embedding and recognition experiments are carried out to test the performance of affine-invariant Fourier based weighted graph(AFWG). Experimental results show that AFWG can preserve the shape feature well and can be applied to shape-related application.