An extremal problem on degree sequences of graphs

Let G = (I_n, E) be the graph of the n-dimensional cube. Namely I_n = {0, 1}ⁿ and [x, y] ε E whenever ||x - y||_I = 1. For A ⊆ I_n and x ε A define h_A(x) = #{y ε I_n / A{[x, y] ε E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A ⊆ I_n of size 2^n-1 we have 1/2ⁿ ∑_xεA √h_A(x) ≥ K for a universal constant K independent of n. We prove a related lower bound for graphs: Let G = (V, E) be a graph with |E| ≥ (k/2). Then ∑_xεV(G) √d(x) ≥ k√k - 1, where d(x) is the degree of x. Equality occurs for the clique on k vertices.