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Eigenspace embeddings of imprimitive association schemes

Janoš Vidali

TL;DR

The paper develops a framework of E_j-eigenspace embeddings to test feasibility of imprimitive, quotient-polynomial graphs and their parameter arrays. By projecting vertex indicators into S_j and enforcing Krein/Eigenmatrix constraints, it systematically rules out several open parameter sets (nonexistence) and proves uniqueness for others (uniqueness). The results demonstrate the practical applicability of spherical representations to partial classifications in the HM databases and illuminate structural features of imprimitive schemes, including explicit automorphism-structure descriptions. The approach blends theoretical embedding criteria with computational tools (SageMath) to advance the understanding of quotient-polynomial graphs and their associated association schemes.

Abstract

For a given symmetric association scheme $\mathcal{A}$ and its eigenspace $S_j$ there exists a mapping of vertices of $\mathcal{A}$ to unit vectors of $S_j$, known as the spherical representation of $\mathcal{A}$ in $S_j$, such that the inner products of these vectors only depend on the relation between the corresponding vertices; furthermore, these inner products only depend on the parameters of $\mathcal{A}$. We consider parameters of imprimitive association schemes listed as open cases in the list of parameters for quotient-polynomial graphs recently published by Herman and Maleki, and study embeddings of their substructures into some eigenspaces consistent with spherical representations of the putative association schemes. Using this, we obtain nonexistence for two parameter sets for $4$-class association schemes and one parameter sets for a $5$-class association scheme passing all previously known feasibility conditions, as well as uniqueness for two parameter sets for $5$-class association schemes.

Eigenspace embeddings of imprimitive association schemes

TL;DR

The paper develops a framework of E_j-eigenspace embeddings to test feasibility of imprimitive, quotient-polynomial graphs and their parameter arrays. By projecting vertex indicators into S_j and enforcing Krein/Eigenmatrix constraints, it systematically rules out several open parameter sets (nonexistence) and proves uniqueness for others (uniqueness). The results demonstrate the practical applicability of spherical representations to partial classifications in the HM databases and illuminate structural features of imprimitive schemes, including explicit automorphism-structure descriptions. The approach blends theoretical embedding criteria with computational tools (SageMath) to advance the understanding of quotient-polynomial graphs and their associated association schemes.

Abstract

For a given symmetric association scheme and its eigenspace there exists a mapping of vertices of to unit vectors of , known as the spherical representation of in , such that the inner products of these vectors only depend on the relation between the corresponding vertices; furthermore, these inner products only depend on the parameters of . We consider parameters of imprimitive association schemes listed as open cases in the list of parameters for quotient-polynomial graphs recently published by Herman and Maleki, and study embeddings of their substructures into some eigenspaces consistent with spherical representations of the putative association schemes. Using this, we obtain nonexistence for two parameter sets for -class association schemes and one parameter sets for a -class association scheme passing all previously known feasibility conditions, as well as uniqueness for two parameter sets for -class association schemes.

Paper Structure

This paper contains 13 sections, 6 theorems, 20 equations, 2 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quotient-polynomial graphs
  4. Eigenspace embeddings of association schemes
  5. A small example
  6. Nonexistence results
  7. QPG with parameter array $[[12, 4, 4, 24], [6, 0, 3; 0, 1; 2]]$
  8. QPG with parameter array $[[8, 8, 4, 24], [1, 0, 2; 2, 1; 1]]$
  9. QPG with parameter array $[[6, 18, 2, 6, 12], [1, 0, 2, 0; 0, 0, 3; 0, 1; 2]]$
  10. Uniqueness results
  11. QPG with parameter array $[[12, 2, 1, 12, 12], [6, 0, 4, 1; 0, 0, 1; 0, 1; 4]]$
  12. QPG with parameter array $[[6, 4, 4, 12, 18], [3, 0, 0, 1; 0, 1, 0; 2, 0; 2]]$
  13. Known constructions and nonexistences

Key Result

Lemma 2

Let $\mathcal{A} = (X, \mathcal{R} = \{R_i \mid i \in \mathcal{I}\})$ be an association scheme with imprimitivity set $\tilde{0}$ and intersection numbers $p^j_{ij} = e-1$ and $p^j_{i'j} = 0$ for some $i \in \tilde{0}$, $j \in \mathcal{I} \setminus \tilde{0}$, and all $i' \in \tilde{0} \setminus \{0

Figures (2)

  • Figure 1: The candidates for the relation scheme $\mathcal{A}^{(3)}$. In each case, the lines are represented with rounded rectangles and form three $\operatorname{GQ}(2, 1)$ geometries; the vertices are implied at intersections of lines. Two distinct vertices are in relation $R^{(3)}_1$ if they are contained in two lines connected by an edge, in relation $R^{(3)}_2$ if they are contained in a common line, in relation $R^{(3)}_3$ if they are contained in distinct lines of the same $\operatorname{GQ}(2, 1)$, and in relation $R^{(3)}_4$ otherwise.
  • Figure 2: The association scheme $\mathcal{A}$. The lines are represented with rounded rectangles and form five $\operatorname{GQ}(2, 1)$ geometries; the vertices are implied at intersections of lines. Two distinct vertices are in relation $R_1$ if they are contained in two lines connected by an edge, in relation $R_2$ if they are contained in a common line, in relation $R_3$ if they are contained in distinct lines of the same $\operatorname{GQ}(2, 1)$, in relation $R_4$ if they are contained in lines of adjacent $\operatorname{GQ}(2, 1)$ geometries not connected by an edge, and in relation $R_5$ otherwise.

Theorems & Definitions (17)

  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 7 more