On the Ubiquity and Utility of Cyclic Schemes

TL;DR

This paper explores cyclic schemes in algebraic graph theory, demonstrating their utility in constructing regular graphs, bipartite graphs, and configurations, and introduces new methods and proofs for these structures.

Contribution

It introduces a unified framework of cyclic schemes for constructing graphs and configurations, providing new constructions and proofs for known combinatorial structures.

Findings

Constructed new bipartite graphs of girth 6 with minimal vertices.

Provided new proofs for existing combinatorial configurations.

Developed a framework linking cyclic schemes to various graph and configuration constructions.

Abstract

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.