We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph G,denotes the robust vertex expansion of G and φ(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 ≥ 2, * É cole polytechnique fédérale de Lausanne,Theorem 4. For any (unknown target) set S ⊆ V , there is a polynomial time randomized algorithm to find a set S with 1. φ(S ′ ) = O(kφ(S)/(ǫφ k (G))) and |S ′ | = O(|S| 1+ǫ ), 2. φ(S ′ ) = O(kφ(S) log(|S|)/φ k (G)) and |S ′ | = O(|S|),