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On Neumaier Cayley graphs

Rhys J. Evans, Sergey Goryainov, Grigory Ryabov, Da Zhao

TL;DR

This work characterizes Neumaier Cayley graphs with spreads given by cosets of a subgroup and provides a practical criterion involving $v=|G|$, $k=|S|$, $s=|H|$, $n=|G:H|$, and $\lambda=s-2+(n-1)m(m-1)/(s-1)$, together with the key containment conditions on $T=S\setminus H^{\#}$. It then constructs infinite families of Neumaier Cayley graphs via relative difference sets (RSDS/RSRDS) and two main constructions, including a novel nexus->1 infinite family, using group-ring calculations and RDS properties. An algorithm for enumerating Neumaier Cayley graphs with a coset spread is developed and implemented in GAP/MAGMA, with computational results up to 64 vertices showing many nexus-1 graphs and a few nexus>1 examples, including a smallest $(64,35,18,4,8)$ instance. The paper advances understanding of how RDS theory yields Neumaier graphs, provides explicit parametric families (notably for even $k$ in a nexus-1+ construction), and delivers a practical enumeration tool that highlights the landscape of such graphs and their regularity properties.

Abstract

In the present paper, we study Neumaier Cayley graphs. First, we give a criterion for a Cayley graph to be a Neumaier graph with a spread given by the cosets of a subgroup. Further, we construct a new infinite family of Neumaier Cayley graphs of unbounded nexus. Finally, we provide an algorithm for enumerating Neumaier Cayley graphs and computational results obtained by this algorithm.

On Neumaier Cayley graphs

TL;DR

This work characterizes Neumaier Cayley graphs with spreads given by cosets of a subgroup and provides a practical criterion involving , , , , and , together with the key containment conditions on . It then constructs infinite families of Neumaier Cayley graphs via relative difference sets (RSDS/RSRDS) and two main constructions, including a novel nexus->1 infinite family, using group-ring calculations and RDS properties. An algorithm for enumerating Neumaier Cayley graphs with a coset spread is developed and implemented in GAP/MAGMA, with computational results up to 64 vertices showing many nexus-1 graphs and a few nexus>1 examples, including a smallest instance. The paper advances understanding of how RDS theory yields Neumaier graphs, provides explicit parametric families (notably for even in a nexus-1+ construction), and delivers a practical enumeration tool that highlights the landscape of such graphs and their regularity properties.

Abstract

In the present paper, we study Neumaier Cayley graphs. First, we give a criterion for a Cayley graph to be a Neumaier graph with a spread given by the cosets of a subgroup. Further, we construct a new infinite family of Neumaier Cayley graphs of unbounded nexus. Finally, we provide an algorithm for enumerating Neumaier Cayley graphs and computational results obtained by this algorithm.
Paper Structure (7 sections, 14 theorems, 25 equations, 3 tables, 1 algorithm)

This paper contains 7 sections, 14 theorems, 25 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main1']}
  3. Relative difference sets
  4. Infinitely many examples from relative difference sets
  5. Construction $1$
  6. Construction $2$
  7. An algorithm for enumeration of Neumaier Cayley graphs with a spread given by the cosets of a subgroup

Key Result

Theorem 1.1

Let $G$ be a group, $H<G$, and $S\subseteq G$ such that $e\notin S$, $S=S^{-1}$, and $H^\#\subseteq S$. Then $\Gamma=\mathop{\mathrm{Cay}}\nolimits(G,S)$ is a Neumaier graph with a regular clique $H$ and parameters $(v,k,\lambda,m,s)$, where $v=|G|$, $k=|S|$, and $s=|H|$, if and only if the followin where $T=S\setminus H^\#$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['main1']}
  • Corollary 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • ...and 14 more