Feature　selection　is　one　of　the　most　important　dimension　reduction　techniques　for　its　efficiency　and　interpretation.　Since　practical　data　in　large　scale　are　usually　collected　without　labels,　and　labeling　these　data　are　dramatically　expensive　and　time-consuming,　unsupervised　feature　selection　has　become　a　ubiquitous　and　challenging　problem.　Without　label　information,　the　fundamental　problem　of　unsupervised　feature　selection　lies　in　how　to　characterize　the　geometry　structure　of　original　feature　space　and　produce　a　faithful　feature　subset,　which　preserves　the　intrinsic　structure　accurately.　In　this　paper,　we　characterize　the　intrinsic　local　structure　by　an　adaptive　reconstruction　graph　and　simultaneously　consider　its　multiconnected-components　(multi-cluster)　structure　by　imposing　a　rank　constraint　on　the　corresponding　Laplacian　matrix.　To　achieve　a　desirable　feature　subset,　we　learn　the　optimal　reconstruction　graph　and　selective　matrix　simultaneously,　instead　of　using　a　predetermined　graph.　We　exploit　an　efficient　alternative　optimization　algorithm　to　solve　the　proposed　challenging　problem,　together　with　the　theoretical　analyses　on　its　convergence　and　computational　complexity.　Finally,　extensive　experiments　on　clustering　task　are　conducted　over　several　benchmark　data　sets　to　verify　the　effectiveness　and　superiority　of　the　proposed　unsupervised　feature　selection　algorithm.