Energy of a graph and Randi\'c index of subgraphs

We give a new inequality between the energy of a graph and a weighted sum over the edges of the graph. Using this inequality we prove that E(G)≥ 2R(H), where E(G) is the energy of a graph G and $R(H)$ is the Randi\'c index of any subgraph of G (not necessarily induced). In particular, this generalizes well-known inequalities E(G)≥ 2R(G) and E(G)≥ 2μ(G) where μ(G) is the matching number. We give other inequalities as applications to this result.