Magnetic and thermal studies of a coordination polymer: bis(glycolato)nickel(ii)

The two-dimensional quadratic lattice magnet, bis(glycolato)cobalt(ii) ([Ni(HOCH2CO2)2]), showed antiferromagnetic ordering at 27.1 K.

Magnetic, thermal, and neutron diffraction studies of a coordination polymer: bis(glycolato)cobalt(ii)

The two-dimensional quadratic lattice magnet, bis(glycolato)cobalt(ii) ([Co(HOCH2CO2)2]), showed antiferromagnetic ordering at 15.2 K and an abrupt increase in magnetisation at H = 22 600 Oe and 2 K, thereby acting as a metamagnet.

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, $0\leq r < m$ and $i+r$ is modulo $m$). Kasteleyn had derived explicit expressions of the number of perfect matchings for $Q_{m,n,0}$ [P.W. Kasteleyn, The statistics of dimers on a 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 expressions of the number of perfect matchings for $Q_{m,n,r}$ by enumerating Pfaffians.