Kernelized Gram matrix (W) constructed from data points (\{x_i\}{i=1}^N) as (W{ij}= k_0( \frac{ | x_i - x_j |^2} {\sigma^2} )) is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth (\sigma), and a common practice called self-tuned kernel adaptively sets a (\sigma_i) at each point (x_i) by the (k)-nearest neighbor (kNN) distance. When (x_i)’s are sampled from a (d)-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator (L_N) to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels (W^{(\alpha)}_{ij} = k_0( \frac{ | x_i - x_j |^2}{ \epsilon \hat{\rho}(x_i) \hat{\rho}(x_j)})/\hat{\rho}(x_i)^\alpha \hat{\rho}(x_j)^\alpha), where (\hat{\rho}) is the estimated bandwidth function {by kNN}, and the limiting operator is also parametrized by (\alpha). When (\alpha = 1), the limiting operator is the weighted manifold Laplacian (\Delta_p). Specifically, we prove the point-wise convergence of (L_N f ) and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a (C^0) consistency for (\hat{\rho}) which bounds the relative estimation error (|\hat{\rho} - \bar{\rho}|/\bar{\rho}) uniformly with high probability, where (\bar{\rho} = p^{-1/d}), and (p) is the data density function. Our theoretical results reveal the advantage of self-tuned kernel over fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of (d) or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data.