English

On commutative association schemes and associated (directed) graphs

Combinatorics 2023-12-04 v2

Abstract

Let M{\cal M} denote the Bose--Mesner algebra of a commutative dd-class association scheme X{\mathfrak X} (not necessarily symmetric), and Γ\Gamma denote a (strongly) connected (directed) graph with adjacency matrix AA. Under the assumption that AA belongs to M{\cal M}, we describe the combinatorial structure of Γ\Gamma. Moreover, we provide an algebraic-combinatorial characterization of Γ\Gamma when AA generates M{\cal M}. Among else, we show that, if X{\mathfrak X} is a commutative 33-class association scheme that is not an amorphic symmetric scheme, then we can always find a (directed) graph Γ\Gamma such that the adjacency matrix AA of Γ\Gamma generates the Bose--Mesner algebra M{\cal M} of X{\mathfrak X}.

Keywords

Cite

@article{arxiv.2307.11680,
  title  = {On commutative association schemes and associated (directed) graphs},
  author = {Giusy Monzillo and Safet Penjić},
  journal= {arXiv preprint arXiv:2307.11680},
  year   = {2023}
}
R2 v1 2026-06-28T11:37:07.127Z