AbstractA quadrilateral cylinder of length m and breadth n is the Cartesian product of am-cycle(with m vertices) and a n-path(with n vertices). Write the vertices of the twocycles on the boundary of the quadrilateral cylinder as x1, x2,


, xm and y1, y2,


, ym,respectively, where xi corresponds to yi(i = 1, 2, . , m). We denote by Qm,n,r, the graphobtained from quadrilateral cylinder of length m and breadth n by adding edges xiyi+r (ris a integer, 0 6 r of the number of perfect matchings for Qm,n,0 [P. W. Kasteleyn, The statistics of dimers ona lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961),1209–1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressionsof the number of perfect matchings for Qm,n,r by enumerating Pfaffians.Keywords: Pfaffian; Perfect matching; Quadratic lattice; Torus...