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The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G, $\\lambda_2 = \\Omega(\\phi^V(G) \\phi(G))$, where $\\phi^V(G)$ denotes the robust vertex expansion of G and $\\phi(G)$ denotes the edge expansion of G. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of G, for all $k \\ge 2$, $\\lambda_2 = \\Omega(\\phi_k(G) \\phi(G) / k)$, where $\\phi_k(G)$ denotes the k-way expansion of G. 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