Nuclei classification provides valuable information for histopathology image analysis. However, the large variations in the appearance of different nuclei types cause difficulties in identifying nuclei. Most neural network based methods are affected by the local receptive field of convolutions, and pay less attention to the spatial distribution of nuclei or the irregular contour shape of a nucleus. In this paper, we first propose a novel polygon-structure feature learning mechanism that transforms a nucleus contour into a sequence of points sampled in order, and employ a recurrent neural network that aggregates the sequential change in distance between key points to obtain learnable shape features. Next, we convert a histopathology image into a graph structure with nuclei as nodes, and build a graph neural network to embed the spatial distribution of nuclei into their representations. To capture the correlations between the categories of nuclei and their surrounding tissue patterns, we further introduce edge features that are defined as the background textures between adjacent nuclei. Lastly, we integrate both polygon and graph structure learning mechanisms into a whole framework that can extract intra and inter-nucleus structural characteristics for nuclei classification. Experimental results show that the proposed framework achieves significant improvements compared to the previous methods. Code and data are made available via https://github.com/lhaof/SENC.