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In this paper we aim to characterize graphs in terms of structural complexities. Our idea is to decompose a graph into substructures of increasing layers, and then to measure the dissimilarity of these substructures using Jensen-Shannon divergence. We commence by identifying a centroid vertex by computing the minimum variance of its shortest path lengths. From the centroid vertex, a family of centroid expansion subgraphs of the graph with increasing layers are constructed. We then compute the depth-based complexity trace of a graph by measuring how the Jensen-Shannon divergence varies with increasing layers of the subgraphs. The required Shannon or von Neumann entropies are computed on the condensed subgraph family of the graph. We perform graph clustering in the principal components space of the complexity trace vector. Experiments on graph datasets abstracted from bioinformatics and image data demonstrate effectiveness and efficiency of the graphs complexity traces.