Equivalence classes of matchings and lattice-square designs

Equivalence classes of matchings and lattice-square designs

We enumerate nonisomorphic lattice-square designs yielded by a conventional construction. Constructed designs are speci3ed by words composed from 3nite-3eld elements. These words are permuted by the isomorphism group in question. The latter group contains a direct-product subgroup, acting, respectively, upon the positions and identities of the 3nite-3eld elements. We review enumeration theory f...

The graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

Sampling Weighted Perfect Matchings on the Square-Octagon Lattice

We consider perfect matchings of the square-octagon lattice, also known as “fortresses” [16]. There is a natural local Markov chain on the set of perfect matchings that is known to be ergodic. However, unlike Markov chains for sampling perfect matchings on the square and hexagonal lattices, corresponding to domino and lozenge tilings, respectively, the seemingly related Markov chain on the squa...

Construction of variety competition cyclic balanced designs on square lattice

A class of cyclic balanced designs with ten test-hills is introduced; these designs can be repeated cyclically any number of times up to the requirement of the experimenter and subject to the availability of experimental material. The designs are balanced with respect to first order nearest neighbours and there are 26 isomorphism classes of such designs. The designs are suitable for de Wit repl...

Some equivalence classes of operators on B(H)