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Some families of graphs, hypergraphs and digraphs defined by systems of equations

Felix Lazebnik, Ye Wang

TL;DR

The survey presents a unifying treatment of graph, hypergraph, and digraph families defined by systems of equations over rings/fields, with a central focus on D(k,q) and Wenger-type graphs. It develops general constructions (BΓ_n, Γ_n, and hypergraphs T and K), analyzes their spectra, automorphisms, connectivity, and decomposition properties, and demonstrates rich applications to generalized polygons, Turán-type extremal problems, and coding/cryptography. By detailing equivalent representations, voltage-lift viewpoints, and extensive specialization results, the paper highlights how algebraic methods yield scalable families of graphs with prescribed girth, regularity, and edge-density, often achieving tight extremal bounds. The work underscores the deep interplay between algebra, combinatorics, and geometry, offering a coherent framework for future exploration of extremal constructions and practical applications in communications and security.

Abstract

The families of graphs defined by a certain type of system of equations over commutative rings have been studied and used since 1990s. This survey presents these families and their applications related to graphs, digraphs, and hypergraphs. Some open problems and conjectures are mentioned.

Some families of graphs, hypergraphs and digraphs defined by systems of equations

TL;DR

The survey presents a unifying treatment of graph, hypergraph, and digraph families defined by systems of equations over rings/fields, with a central focus on D(k,q) and Wenger-type graphs. It develops general constructions (BΓ_n, Γ_n, and hypergraphs T and K), analyzes their spectra, automorphisms, connectivity, and decomposition properties, and demonstrates rich applications to generalized polygons, Turán-type extremal problems, and coding/cryptography. By detailing equivalent representations, voltage-lift viewpoints, and extensive specialization results, the paper highlights how algebraic methods yield scalable families of graphs with prescribed girth, regularity, and edge-density, often achieving tight extremal bounds. The work underscores the deep interplay between algebra, combinatorics, and geometry, offering a coherent framework for future exploration of extremal constructions and practical applications in communications and security.

Abstract

The families of graphs defined by a certain type of system of equations over commutative rings have been studied and used since 1990s. This survey presents these families and their applications related to graphs, digraphs, and hypergraphs. Some open problems and conjectures are mentioned.

Paper Structure

This paper contains 56 sections, 53 theorems, 114 equations, 11 tables.

Table of Contents

  1. Introduction
  2. Main constructions
  3. Bipartite version
  4. Ordinary version
  5. Uniform $k$-partite hypergraphs
  6. General properties of graphs $B\Gamma_n$, $\Gamma_n$ and hypergraphs ${\mathcal{T}}$, ${\mathcal{K}}$
  7. Degrees and neighbor-complete colorings
  8. Special induced subgraphs
  9. Covers and lifts
  10. Embedded spectra
  11. Edge-decomposition of $K_n$ and $K_{m,m}$
  12. Edge-decomposition of complete $k$-partite $r$-graphs and complete $r$-graphs
  13. Specializations of general constructions and their applications
  14. Generalized polygons
  15. Turán-type extremal problems
  16. ...and 41 more sections

Key Result

Proposition 1

The graphs $B\Gamma(R;f_2,\ldots, f_n)$ and $B\Gamma(R;\overline{f_2},\ldots, \overline{f_n})$ are isomorphic, an explicit isomorphism being given by $\varphi: (a)\leftrightarrow [a]$.

Theorems & Definitions (71)

  • Proposition 1
  • Theorem 1
  • proof
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 61 more