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Graph convolutional network (GCN) is now an effective tool to deal with non-Euclidean data, such as social networks in social behavior analysis, molecular structure analysis in the field of chemistry, and skeleton-based action recognition. Graph convolutional kernel is one of the most significant factors in GCN to extract nodes' feature, and some improvements of it have reached promising performance theoretically and experimentally. However, there is limited research about how exactly different data types and graph structures influence the performance of these kernels. Most existing methods used an adaptive convolutional kernel to deal with a given graph structure, which still not reveals the internal reasons. In this paper, we started from theoretical analysis of the spectral graph and studied the properties of existing graph convolutional kernels. While taking some designed datasets with specific parameters into consideration, we revealed the self-smoothing phenomenon of convolutional kernels. After that, we proposed the Poisson kernel that can avoid self-smoothing without training any adaptive kernel. Experimental results demonstrate that our Poisson kernel not only works well on the benchmark dataset where state-of-the-art methods work fine, but also is evidently superior to them in synthetic datasets.