Spanning trees and dimer problem on the Cairo pentagonal lattice

The Cairo pentagonal lattice is the dual lattice of the (32.4.3.4) lattice. In this work, we obtain explicit expression of the number of spanning trees of the Cairo pentagonal lattice with toroidal boundary condition, particularly, there is a constant difference (not one) of the number of spanning trees between the (32.4.3.4) lattice and the Cairo pentagonal lattice with toroidal boundary condition. We also obtain the asymptotic growth constant and the dimer entropy of the Cairo pentagonal lattice with toroidal boundary condition.