In general, deep neural network (DNN) pruning methods fall into two categories: 1) weight-based deterministic constraints and 2) probabilistic frameworks. While each approach has its merits and limitations, there are a set of common practical issues such as trial-and-error to analyze sensitivity and hyper-parameters to prune DNNs, which plague them both. In this work, we propose a new single-shot, fully automated pruning algorithm called slimming neural networks using adaptive connectivity scores (SNACS). Our proposed approach combines a probabilistic pruning framework with constraints on the underlying weight matrices, via a novel connectivity measure, at multiple levels to capitalize on the strengths of both approaches while solving their deficiencies. In SNACS, we propose a fast hash-based estimator of adaptive conditional mutual information (ACMI), that uses a weight-based scaling criterion, to evaluate the connectivity between filters and prune unimportant ones. To automatically determine the limit up to which a layer can be pruned, we propose a set of operating constraints that jointly define the upper pruning percentage limits across all the layers in a deep network. Finally, we define a novel sensitivity criterion for filters that measures the strength of their contributions to the succeeding layer and highlights critical filters that need to be completely protected from pruning. Through our experimental validation, we show that SNACS is faster by over 17× the nearest comparable method and is the state-of-the-art single-shot pruning method across four standard Dataset-DNN pruning benchmarks: CIFAR10-VGG16, CIFAR10-ResNet56, CIFAR10-MobileNetv2, and ILSVRC2012-ResNet50.