The dependence of graph energy on network structure

In this paper, we aim at investigating how the energy of a graph depends upon its underlying topological structure for regular and sparse scale free networks. Firstly, the spectra and energies of some simple regular graphs are calculated exactly and an exact expression is derived for the eigenvalues of adjacency matrix of regular graphs with degree k being given by k = 2^a (a = 1, 2, 3,...). It is also found that a graph with k being about 0.8N owns the largest energy for the regular graphs with the same size and the same generating method used in this paper. Furthermore, we investigate the energy of sparse scale-free networks with different average degree < k > and degree distribution exponent γ. While γ is specified, the energy is a power-law function of < k > with exponent being about 0.5. And while < k > is fixed, energy will be obviously proportional to γ. Otherwise, we also find that the energy is a power- law function of the variance of degree sequence with exponent weakly depending on the size of network. Interestingly, while both < k > and γ are specified, there will be a terrific linear fit to the relationship between energy and the size of system. (paper)