We introduce a novel regularization approach for deep learning that incorporates and respects the underlying graphical structure of the neural network. Existing regularization methods often focus on dropping/penalizing weights in a global manner that ignores the connectivity structures of the neural network. We propose to use the Fiedler value of the neural network's underlying graph as a tool for regularization. We provide theoretical support for this approach via Spectral Graph Theory. We demonstrate the convexity of this penalty and provide an approximate, variational approach for fast computation in practical training of neural networks. We provide bounds on such approximations. We provide an alternative but equivalent formulation of this framework in the form of a structurally weighted L1 penalty, thus linking our approach to sparsity induction. We trained neural networks on various datasets to compare Fiedler Regularization with traditional regularization methods such as Dropout and weight decay. Results demonstrate the efficacy of Fiedler Regularization.