Spectral clustering is a popular approach for clustering undirected graphs, but its extension to directed graphs (digraphs) is much more challenging. A typical workaround is to naively symmetrize the adjacency matrix of the directed graph, which can however lead to discarding valuable information carried by edge directionality. In this paper, we present a generalized spectral clustering framework that can address both directed and undirected graphs. Our approach is based on the spectral relaxation of a new functional that we introduce as the generalized Dirichlet energy of a graph function, with respect to an arbitrary positive regularizing measure on the graph edges. We also propose a practical parametrization of the regularizing measure constructed from the iterated powers of the natural random walk on the graph. We present theoretical arguments to explain the efficiency of our framework in the challenging setting of unbalanced classes. Experiments using directed K-NN graphs constructed from real datasets show that our graph partitioning method performs consistently well in all cases, while outperforming existing approaches in most of them.