Abstract
The graph energy, as a graph invariant, contains very important structural information about the graph, its subgraphs, and ingredient segments. Estrada and Benzi (2017) have shown that the graph energy can be extracted from the weighted sum of the traces of the even powers of its adjacency matrix. Based on this kind of representation, they have reported newer and more precise bounds of the energy as the sum of fragments’ contributions such as Cn (cycles of length n), Pn (path graph), Sn (star graph), Dn (diamond graph), F (a subgraph containing a square with a pendant vertex), and H (a subgraph containing two triangles with a shared vertex). In this paper, inspired by the work of Estrada and Benzi, we first introduce a general formula for calculating the contribution of subgraphs to the total energy of the graph. We also compute the boundaries of the contributions of the above subgraphs and some of the subgraphs that appear for the first time in the higher traces of the adjacency matrix of the graph. Further, we calculate the upper bound of the graph energy for cyclic and fullerene graphs with higher accuracy.
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