On the Ubiquity and Utility of Cyclic Schemes

Let $k,l,m,n$, and $\mu$ be positive integers. A $\mathbb{Z}_\mu$--{\it scheme of valency} $(k,l)$ and {\it order} $(m,n)$ is a $m \times n$ array $(S_{ij})$ of subsets $S_{ij} \subseteq \mathbb{Z}_\mu$ such that for each row and column one has $\sum_{j=1}^n |S_{ij}| = k$ and $\sum_{i=1}^m |S_{ij}| = l$, respectively. Any such scheme is an algebraic equivalent of a $(k,l)$-semi-regular bipartite voltage graph with $n$ and $m$ vertices in the bipartition sets and voltages coming from the cyclic group $\mathbb{Z}_\mu$. We are interested in the subclass of $\mathbb{Z}_\mu$--schemes that are characterized by the property $a - b + c - d\; \not \equiv \;0$ (mod $\mu$) for all $a \in S_{ij}$, $b \in S_{ih}$, $c \in S_{gh}$, and $d \in S_{gj}$ where $i,g \in {1,...,m}$ and $j,h \in {1,...,n}$ need not be distinct. These $\mathbb{Z}_\mu$--schemes can be used to represent adjacency matrices of regular graphs of girth $\ge 5$ and semi-regular bipartite graphs of girth $\ge 6$. For suitable $\rho, \sigma \in \mathbb{N}$ with $\rho k = \sigma l$, they also represent incidence matrices for polycyclic $(\rho \mu_k, \sigma \mu_l)$ configurations and, in particular, for all known Desarguesian elliptic semiplanes. Partial projective closures yield {\it mixed $\mathbb{Z}_\mu$-schemes}, which allow new constructions for Kr\v{c}adinac's sporadic configuration of type $(34_6)$ and Balbuena's bipartite $(q-1)$-regular graphs of girth 6 on as few as $2(q^2-q-2)$ vertices, with $q$ ranging over prime powers. Besides some new results, this survey essentially furnishes new proofs in terms of (mixed) $\mathbb{Z}_\mu$--schemes for ad-hoc constructions used thus far.