  For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$, the energy of the graph
is defined by $E(G)=\sum_{j=1}^{n}|\lambda_{j}|$. Myriads of papers have been
published in the mathematical and chemistry literature about properties of this
graph invariant due to its connection with the energy of (bipartite) conjugated
molecules. However, a structural interpretation of this concept in terms of the
contributions of even and odd walks, and consequently on the contribution of
subgraphs, is not yet known. Here, we find such interpretation and prove that
the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the
traces of even powers of the adjacency matrix. We then use such result to find
bounds for the energy in terms of subgraphs contributing to it. The new bounds
are studied for some specific simple graphs, such as cycles and fullerenes. We
observe that including contributions from subgraphs of sizes not bigger than 6
improves some of the best known bounds for the energy, and more importantly
gives insights about the contributions of specific subgraphs to the energy of
these graphs.
