Abstarct: The concept of graph energy arose in the context of the study of conjugated hydrocarbons in chemistry, and it used to approximate the total π-electron energy of molecules. It is defined as the sum of the absolute values of eigenvalues of adjacency matrix assigned to a graph G. In this thesis, first we provide a structural interpretation of the graph energy using the concept of matrix functions. In particular, we prove that the graph energy is given by the sum of the traces of the even powers of the adjacency matrix weighted in a specific way. Using this new representation, we find new bounds for the energy as sums of contributions of subgraphs. Second, we investigate how the energy of a graph depends upon its underlying topological structure for regular and sparse scale free networks. Moreover, spectra and energies of some classes of graphs are calculated.