On the minimal energy of graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph which is used to approximate the total π-electron energy of the molecule. In this paper, we determine the (n,e)-graphs with minimal energy for e=n+1 and n+2, which is giving a complete solution to the conjecture for e=n+1 and e=n+2 proposed by Caporossi et al. in [4]. Moreover, we determine the graphs with the minimal and second-minimal energies for n−1≤e≤3n2−3, and the unique graph with minimal energy for 3n−52≤e≤2n−4 among all quasi-trees with n vertices and e edges, respectively.