The triangular kagomé lattices revisited

• It is reported that the triangular kagomé lattice is the line graph of the 3.12.12 lattice.
• The enumerative and asymptotic formulae of the number of spanning trees of the triangular kagomé lattice are obtained.
• We obtain the Kirchhoff index of the triangular kagomé lattice.
• We derive the formula of the energy of the triangular kagomé lattice.

The dimer problem, Ising spins and bond percolation on the triangular kagomé lattice have been studied extensively by physicists. In this paper, based on the fact the triangular kagomé lattice with toroidal boundary condition can be regarded as the line graph of 3.12.12 lattice with toroidal boundary condition, we derive the formulae of the number of spanning trees, the energy, and the Kirchhoff index of the triangular kagomé lattice with toroidal boundary condition.