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Clustering techniques attempt to group objects with similar properties into a cluster. Clustering the nodes of an attributed graph, in which each node is associated with a set of feature attributes, has attracted significant attention. Graph convolutional networks (GCNs) represent an effective approach for integrating the two complementary factors of node attributes and structural information for attributed graph clustering. However, oversmoothing of GCNs produces indistinguishable representations of nodes, such that the nodes in a graph tend to be grouped into fewer clusters, and poses a challenge due to the resulting performance drop. In this study, we propose a smoothness sensor for attributed graph clustering based on adaptive smoothness-transition graph convolutions, which senses the smoothness of a graph and adaptively terminates the current convolution once the smoothness is saturated to prevent oversmoothing. Furthermore, as an alternative to graph-level smoothness, a novel fine-gained node-wise level assessment of smoothness is proposed, in which smoothness is computed in accordance with the neighborhood conditions of a given node at a certain order of graph convolution. In addition, a self-supervision criterion is designed considering both the tightness within clusters and the separation between clusters to guide the whole neural network training process. Experiments show that the proposed methods significantly outperform 12 other state-of-the-art baselines in terms of three different metrics across four benchmark datasets. In addition, an extensive study reveals the reasons for their effectiveness and efficiency.