Clustering is one of the most useful tools for high-dimensional analysis, e.g., for microarray data. It becomes challenging in presence of a large number of noise variables, which may mask underlying clustering structures. Therefore, noise removal through variable selection is necessary. One effective way is regularization for simultaneous parameter estimation and variable selection in model-based clustering. However, existing methods focus on regularizing the mean parameters representing centers of clusters, ignoring dependencies among variables within clusters, leading to incorrect orientations or shapes of the resulting clusters. In this article, we propose a regularized Gaussian mixture model permitting a treatment of general covariance matrices, taking various dependencies into account. At the same time, this approach shrinks the means and covariance matrices, achieving better clustering and variable selection. To overcome one technical challenge in estimating possibly large covariance matrices, we derive an E-M algorithm utilizing the graphical lasso (Friedman et al 2007) for parameter estimation. Numerical examples, including applications to microarray gene expression data, demonstrate the utility of the proposed method.