Torsion-weighted spanning acycle entropy in cubical lattices and Mahler measures

We compute the eigenvalues of up-Laplacians on cubical lattices and derive the torsion-weighted count of spanning acycles in cubical lattices by using the matrix-tree theorem for simplicial/cell complexes. As a corollary, we show that the torsion-weighted spanning acycle entropy of a cubical lattice defined on \(\mathbb {Z}^b \times \prod _{j=b+1}^q \{0,1,\dots , n_j\}\) ${\mathbb{Z}}^b\times {\prod }_{j=b+1}^q\{0,1,\cdots ,n_j\}$ is expressed as a linear combination of logarithmic Mahler measures.