The energy of a type of lattice

The energy of a simple graph GG, denoted by E(G)E(G), is defined as the sum of the absolute values of eigenvalues of GG. Let Gf(m,n)Gf(m,n) (resp., Gc(m,n)Gc(m,n) and Gt(m,n)Gt(m,n)) be the lattice obtained from an m×nm×n square lattice with free boundary condition (resp., cylindrical boundary condition and the toroidal boundary condition) by adding two diagonal edges to each square. In this work, we obtain the formulae for the energies of Gf(m,n)Gf(m,n), Gc(m,n)Gc(m,n), and Gt(m,n)Gt(m,n). The asymptotic energies of Gf(m,n)Gf(m,n), Gc(m,n)Gc(m,n) and Gt(m,n)Gt(m,n) are also given.