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Deza Cayley graphs from difference sets

Grigory Ryabov

TL;DR

The paper constructs new Deza Cayley graphs over generalized dihedral groups by exploiting difference sets and relative difference sets, yielding two infinite families of strictly Deza graphs with explicit parameter patterns. It employs group-ring methods to verify the Deza property and develops two main constructions that produce graphs with diameter 2 or strong regularity under specified parameter conditions. It further shows how to generate additional Deza Cayley graphs through standard graph operations, expanding the known catalog of graphs in this class. The work advances understanding of how combinatorial designs translate into Deza graph families and outlines open questions about discovering further DS/RDS-based examples.

Abstract

In this note, we provide several constructions of Deza Cayley graphs over groups having a generalized dihedral subgroup. These constructions are based on a usage of (relative) difference sets.

Deza Cayley graphs from difference sets

TL;DR

The paper constructs new Deza Cayley graphs over generalized dihedral groups by exploiting difference sets and relative difference sets, yielding two infinite families of strictly Deza graphs with explicit parameter patterns. It employs group-ring methods to verify the Deza property and develops two main constructions that produce graphs with diameter 2 or strong regularity under specified parameter conditions. It further shows how to generate additional Deza Cayley graphs through standard graph operations, expanding the known catalog of graphs in this class. The work advances understanding of how combinatorial designs translate into Deza graph families and outlines open questions about discovering further DS/RDS-based examples.

Abstract

In this note, we provide several constructions of Deza Cayley graphs over groups having a generalized dihedral subgroup. These constructions are based on a usage of (relative) difference sets.

Paper Structure

This paper contains 6 sections, 9 theorems, 25 equations, 1 table.

Table of Contents

  1. Introduction
  2. Difference sets
  3. Group ring
  4. Constructions
  5. More Deza graphs
  6. Concluding remarks

Key Result

Theorem 1.1

There exist strictly Deza Cayley graphs with parameters and over the generalized dihedral groups associated with the groups $\mathbb{Z}_4^{k-1}\times \mathbb{Z}_2$ and $\mathbb{Z}_{\frac{q^2-1}{2}}$, respectively, where $k\geq 3$ and $q\geq 5$ is an odd prime power.

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 7 more