Determinants of grids, tori, cylinders and Mobius ladders

Recently, Bien [A. Bien, The problem of singularity for planar grids, Discrete Math. 311 (2011) 921-931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box products of paths, Discrete Math. 312 (2012) 1844-1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, that of a cylinder, and that of a Mobius ladder. (C) 2013 Elsevier B.V. All rights reserved.