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Clustering based on the random walk operator has been proven effective for undirected graphs, but its generalization to directed graphs (digraphs) is much more challenging. Although the random walk operator is well-defined for digraphs, in most cases such graphs are not strongly connected, and hence the associated random walks are not irreducible, which is a crucial property for clustering that exists naturally in the undirected setting. To remedy this, the usual workaround is to either naively symmetrize the adjacency matrix or to replace the natural random walk operator by the teleporting random walk operator, but this can lead to the loss of valuable information carried by edge directionality. In this paper, we introduce a new clustering framework, the Parametrized Random Walk Diffusion Kernel Clustering (P-RWDKC), which is suitable for handling both directed and undirected graphs. Our framework is based on the diffusion geometry and the generalized spectral clustering framework. Accordingly, we propose an algorithm that automatically reveals the cluster structure at a given scale, by considering the random walk dynamics associated with a parametrized kernel operator, and by estimating its critical diffusion time. Experiments on $K$-NN graphs constructed from real-world datasets and real-world graphs show that our clustering approach performs well in all tested cases, and outperforms existing approaches in most of them.