What is the meaning of the graph energy after all?

Abstract

For a simple graph G=V,E with eigenvalues of the adjacency matrix λ1≥λ2≥⋯≥λn, the energy of the graph is defined by EG=∑j=1n|λj|. 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 an 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 bound 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 energy, and more importantly gives insights about the contributions of specific subgraphs to the energy of these graphs.