Asymptotic energy of lattices

The energy of a simple graph GG arising in chemical physics, denoted by E(G)E(G), is defined as the sum of the absolute values of eigenvalues of GG. As the dimer problem and spanning trees problem in statistical physics, in this paper we propose the energy per vertex problem for lattice systems. In general for a type of lattice in statistical physics, to compute the entropy constant with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness and may have different solutions. We show that the energy per vertex of plane lattices is independent of the toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions. In particular, the asymptotic formulae of energies of the triangular, 33.4233.42, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are obtained explicitly.