In　the　graph　signal　processing　(GSP)　literature,　graph　Laplacian　regularizer　(GLR)　was　used　for　signal　restoration　to　promote　piecewise　smooth　/　constant　reconstruction　with　respect　to　an　underlying　graph.　However,　for　signals　slowly　varying　across　graph　kernels,　GLR　suffers　from　an　undesirable　＇staircase＇　effect.　In　this　paper,　focusing　on　manifold　graphs　-　collections　of　uniform　discrete　samples　on　low-dimensional　continuous　manifolds　-　we　generalize　GLR　to　gradient　graph　Laplacian　regularizer　(GGLR)　that　promotes　planar　/　piecewise　planar　(PWP)　signal　reconstruction.　Specifically,　for　a　graph　endowed　with　sampling　coordinates　(e.g.,　2D　images,　3D　point　clouds),　we　first　define　a　gradient　operator,　using　which　we　construct　a　gradient　graph　for　nodes＇　gradients　in　the　sampling　manifold　space.　This　maps　to　a　gradient-induced　nodal　graph　(GNG)　and　a　positive　semi-definite　(PSD)　Laplacian　matrix　with　planar　signals　as　the　$0$　frequencies.　For　manifold　graphs　without　explicit　sampling　coordinates,　we　propose　a　graph　embedding　method　to　obtain　node　coordinates　via　fast　eigenvector　computation.　We　derive　the　means-square-error　minimizing　weight　parameter　for　GGLR　efficiently,　trading　off　bias　and　variance　of　the　signal　estimate.　Experimental　results　show　that　GGLR　outperformed　previous　graph　signal　priors　like　GLR　and　graph　total　variation　(GTV)　in　a　range　of　graph　signal　restoration　tasks.　　©　1991-2012　IEEE.