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Lower bound for the second smallest eigenvalue of directed rooted graph Laplacian

A lower bound for the second smallest eigenvalue (SSE) of the unweighted Laplacian for an N-vertex directed rooted graph is obtained by obtaining the supremum of the scrambling constant of the (N-1)-th power of the corresponding adjacency matrix. This supremum is actually achieved by the ”N-layer Complete Graph” (NCG) defined in this paper, which implies that for directed rooted graphs that is unweighted for its directed edges, NCGs have the least connective topology in the sense of scrambling constant.