The energy of a graph and its size dependence. A Monte Carlo approach

The energy E of a graph is the sum of absolute values of graph eigenvalues and is a proper generalization of the total π-electron energy of conjugated hydrocarbons. The dependence of E on the number n of vertices and the number m of edges is examined by means of a Monte Carlo approach. It is shown that, for a fixed and sufficiently large value of n, E is, on average, an increasing function of m only for relatively small values of m and reaches a maximum for graphs in which about two-thirds of the vertex pairs are adjacent. This maximal energy increases as n1.4.