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Neumaier graphs from cyclotomy with small coherent rank

Gary R. W. Greaves, Zhao Kuang Tan

Abstract

Using cyclotomy, we construct a new infinite family of Neumaier graphs that includes infinitely many strongly regular graphs. Notably, this family conjecturally contains infinitely many graphs with coherent rank $6$. Our construction also provides the first known examples that answer a question posed by Evans, Goryainov, and Panasenko regarding the existence of Neumaier graphs whose nexus is not a power of $2$. In addition, we show that a construction of Greaves and Koolen yields an infinite family of Neumaier graphs with coherent rank $6$.

Neumaier graphs from cyclotomy with small coherent rank

Abstract

Using cyclotomy, we construct a new infinite family of Neumaier graphs that includes infinitely many strongly regular graphs. Notably, this family conjecturally contains infinitely many graphs with coherent rank . Our construction also provides the first known examples that answer a question posed by Evans, Goryainov, and Panasenko regarding the existence of Neumaier graphs whose nexus is not a power of . In addition, we show that a construction of Greaves and Koolen yields an infinite family of Neumaier graphs with coherent rank .

Paper Structure

This paper contains 19 sections, 30 theorems, 56 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Coherent closure of a graph
  3. Coherent algebras and configurations
  4. Coherent closure and coherent rank
  5. Small coherent rank
  6. An infinite family with coherent rank 6
  7. A new family of Neumaier graphs from cyclotomy
  8. Cyclotomic numbers
  9. Schur rings and Cayley graphs
  10. The general construction
  11. Coherent rank 6 and beyond
  12. More strongly regular graphs
  13. Coherent rank 6
  14. Coherent rank 7
  15. Beyond coherent rank 7
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\mathcal{A}$ be a coherent algebra and let $A \in \mathcal{A}$. For $b \in \mathbb C$, define the matrix $B$ such that Then $B \in \mathcal{A}$.

Theorems & Definitions (48)

  • Lemma 2.1: cc
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.5: cf. abiad2021neumaier
  • Theorem 2.7
  • proof
  • Proposition 2.8
  • proof
  • Theorem 2.9: cf. GreavesKoolen2
  • ...and 38 more