On the ubiquity and utility of cyclic schemes

Let k, l,m, n, and μ be positive integers. A Zμ-scheme of valency (k, l) and order (m,n) is an m × n array (Sij) of subsets Sij ⊆ Zμ such that for each row and column one has ∑n j=1 |Sij | = k and ∑m i=1 |Sij | = l, respectively. Any such scheme is an algebraic equivalent of a (k, l)-semiregular bipartite voltage graph with n and m vertices in the bipartition sets and voltages coming from the cyclic group Zμ. We are interested in the subclass of Zμ-schemes that are characterized by the property a− b+ c−d ≡ 0 (mod μ) for all a ∈ Sij, b ∈ Sih, c ∈ Sgh, and d ∈ Sgj where i, g ∈ {1, . . . ,m} and j, h ∈ {1, . . . , n} need not be distinct. These Zμ-schemes can be used to represent adjacency matrices of regular graphs of girth ∗ This research was was carried out within the activity of INdAM-GNSAGA and supported by the Italian Ministry MIUR. † Corresponding author 96 M. ABREU, M.J. FUNK, D. LABBATE AND V. NAPOLITANO ≥ 5 and semi-regular bipartite graphs of girth ≥ 6. For suitable ρ, σ ∈ N with ρk = σl, they also represent incidence matrices for polycyclic (ρμk, σμl) configurations and, in particular, for all known Desarguesian elliptic semiplanes. Partial projective closures yield mixed Zμ-schemes , which allow new constructions for Krčadinac’s sporadic configuration of type (346) and Balbuena’s bipartite (q − 1)-regular graphs of girth 6 on as few as 2(q−q−2) vertices, with q ranging over prime powers. Besides some new results, this survey essentially furnishes new proofs in terms of (mixed) Zμ-schemes for ad hoc constructions used thus far. 1 Zμ-Schemes and Cyclic Voltage Graphs Preliminary note. This paper deals with constructions in some classes of (0, 1)matrices, which turn up as incidence matrices of configurations or adjacency matrices of graphs. Even if basic notions and notations seem to be generally known and widely used, misunderstandings can arise since precise formal definitions vary slightly from author to author. So one might be tempted to fix every notion to the least detail, at the risk of distracting the reader’s attention from the essentially new concepts. To overcome this dilemma, the reader will find a synopsis on (0, 1)-matrices, graphs, and configurations in Section 9. Notions defined in the synopsis are set up in italics at their very first appearance in the paper. Definition 1.1 A Zμ-scheme of order (m,n) is an m × n array M (μ) = (Sij) of subsets Sij ⊆ Zμ. The Zμ-scheme M (μ) has valency (k, l) if, for each row and column, the sums of the cardinalities of the entries have constant values k and l, i.e.