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 $\mathcal{E}(G)\geq 2R(H)$, where $\mathcal{E}(G)$ 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 $\mathcal{E}(G)\geq 2R(G)$ and $\mathcal{E}(G)\geq 2\mu(G)$ where $\mu(G)$ is the matching number. We give other inequalities as applications to this result.