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m-distance-regular graphs and their relation to multivariate P-polynomial association schemes

Pierre-Antoine Bernard, Nicolas Crampe, Luc Vinet, Meri Zaimi, Xiaohong Zhang

TL;DR

The notion of $m$-distance-regular graph is defined and shown to give a graph interpretation of the multivariate $P$-polynomial association schemes.

Abstract

An association scheme is $P$-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate $P$-polynomial association schemes of type $(α,β)$ were introduced by Bernard et al., and multivariate $P$-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of $m$-distance-regular graph is defined and shown to give a graph interpretation of the multivariate $P$-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate $P$-polynomial association schemes and $m$-distance-regular graphs are also considered. In particular, bivariate $P$-polynomial schemes of type $(α, β)$ are discussed, and their connection to 2-distance-regular graphs is established.

m-distance-regular graphs and their relation to multivariate P-polynomial association schemes

TL;DR

The notion of -distance-regular graph is defined and shown to give a graph interpretation of the multivariate -polynomial association schemes.

Abstract

An association scheme is -polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate -polynomial association schemes of type were introduced by Bernard et al., and multivariate -polynomial association schemes were later defined by Bannai et al. In this paper, the notion of -distance-regular graph is defined and shown to give a graph interpretation of the multivariate -polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate -polynomial association schemes and -distance-regular graphs are also considered. In particular, bivariate -polynomial schemes of type are discussed, and their connection to 2-distance-regular graphs is established.
Paper Structure (14 sections, 22 theorems, 85 equations, 3 figures)

This paper contains 14 sections, 22 theorems, 85 equations, 3 figures.

Table of Contents

  1. Background
  2. Multivariate $P$-polynomial association schemes
  3. $m$-distance-regular graphs
  4. Examples
  5. Cartesian product of distance-regular graphs
  6. Hamming graphs and the symmetrization of association schemes
  7. 24-cell
  8. Refinements and additional conditions
  9. Partial order refinement
  10. Conditions on the domain and its boundary
  11. Bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$
  12. Generalized 24-cell
  13. Outlook
  14. Complementary proofs

Key Result

Proposition 2.4

(Proposition 2.15 of bannai2023multivariate) Let $\mathcal{D}\subset \mathbb{N}^m$ such that $e_1, e_2, \dots, e_m \in \mathcal{D}$ and $\mathcal{Z} = \{A_n \ | \ n \in \mathcal{D}\}$ be a commutative association scheme. Then the following two statements are equivalent:

Figures (3)

  • Figure 1: Cartesian product of the cycle graphs $C_{14}$ and $C_{9}$. The result is a two-dimensional toroidal square lattice, which is $2$-distance-regular with respect to the blue and red colouring of its edges and any monomial order.
  • Figure 2: Hamming graphs $H(2,4)$ on the left and $H(3,4)$ on the right. They are $2$-distance-regular with respect to the blue and red partition of the edges and the deg-lex order. This is the partition obtained using the symmetrizations $\mathcal{S}^2(\mathcal{Z})$ and $\mathcal{S}^3(\mathcal{Z})$ of $\mathcal{Z} = \{I \otimes I, I \otimes \sigma_x + \sigma_x \otimes I, \sigma_x \otimes \sigma_x \}$.
  • Figure 3: Two domains on which the generalized 24-cell association schemes are bivariate $P$-polynomial of type $(\alpha,\beta)$, with $0\leq \alpha \leq 1$, $0\leq \beta < 1$ in the first case and $\frac{1}{2} \leq \alpha < 1$, $0\leq \beta < 1$ in the second case.

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 45 more