Graph-based　clustering　learns　underlying　data　representation　by　employing　topological　graph　structure.　Recently,　Graph　Convolutional　Network　(GCN)-based　clustering　methods　have　accumulated　great　attentions　and　achieved　great　performance.　Its　performance　is　seriously　determined　by　the　quality　of　pre-provided　graph,　which　is　usually　constructed　by　predefined　model　(such　as　k-Nearest-Neighbor).　However,　the　graph　may　be　inaccurate　due　to　the　noises　and　fixed　graph　limits　flexibility　of　model　learning.　In　this　paper,　we　propose　a　Robust　Graph　Convolutional　Clustering　(RGCC)　method,　which　adaptively　learns　a　clean　and　accurate　graph　from　original　graph.　Specifically,　adaptive　graph　with　low-rank　and　sparse　structures　be　learned　during　the　optimization　process,　which　can　better　encode　structural　information　of　data　than　fixed　graph.　Then,　to　explore　the　local　connectivity　of　data,　graph　Laplacian　constraint　is　introduced.　Thus,　optimal　graph　relationships　and　discriminative　representation　of　data　could　be　simultaneously　learned,　which　improves　the　flexibility　of　the　RGCC　model.　By　designing　a　self-supervised　clustering　module,　it　can　self-supervise　the　node　representations　learning　and　thus　explore　the　better　clustering　structure.　Experimental　results　on　several　benchmark　databases　reveal　the　superiority　of　the　proposed　RGCC　approach.