Graph　Convolutional　Network　(GCN)　has　drawn　widespread　attention　in　data　mining　on　graphs　due　to　its　outstanding　performance　and　rigor　theoretical　guarantee.　However,　some　recent　studies　have　revealed　that　GCN-based　methods　may　mine　latent　information　insufficiently　owing　to　the　underutilization　of　the　feature　space.　Besides,　the　unlearnable　topology　also　significantly　imperils　the　performance　of　GCN-based　methods.　In　this　paper,　we　conduct　experiments　to　investigate　these　issues,　finding　that　GCN　does　not　fully　consider　the　potential　structure　in　the　feature　space,　and　a　fixed　topology　deteriorates　the　robustness　of　GCN.　Thus,　it　is　desired　to　distill　node　features　and　establish　a　learnable　graph.　Motivated　by　this　goal,　we　propose　a　framework　dubbed　Graph　Convolutional　Network　with　elastic　topology　(GCNet1).　With　the　analysis　of　the　optimization　for　the　proposed　flexible　Laplacian　embedding,　GCNet　is　naturally　constructed　by　alternative　graph　convolutional　layers　and　adaptive　topology　learning　layers.　GCNet　aims　to　deeply　explore　the　feature　space　and　employ　the　mined　information　to　construct　a　learnable　topology,　which　leads　to　a　more　robust　graph　representation.　In　addition,　a　set　-level　orthogonal　loss　is　utilized　to　meet　the　orthogonal　constraint　required　by　the　flexible　Laplacian　embedding　and　promote　better　class　separability.　Moreover,　comprehensive　experiments　indicate　that　GCNet　achieves　remarkable　performance　and　generalization　on　several　real　-world　datasets.