Enumeration of Perfect Matchings of a Type of Quadratic Lattice on the Torus

A quadrilateral cylinder of length $m$ and breadth $n$ is the Cartesian product of a $m$-cycle(with $m$ vertices) and a $n$-path(with $n$ vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as $x_1,x_2,\cdots,x_m$ and $y_1,y_2,\cdots ,y_m$, respectively, where $x_i$ corresponds to $y_i(i=1,2,\dots, m)$. We denote by $Q_{m,n,r}$, the graph obtained from quadrilateral cylinder of length $m$ and breadth $n$ by adding edges $x_iy_{i+r}$ ($r$ is a integer, $0\leq r