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arXiv · 2307.11680

On commutative association schemes and associated (directed) graphs

Giusy Monzillo·Safet Penjić
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Abstract

Let ${\cal M}$ denote the Bose--Mesner algebra of a commutative $d$-class association scheme ${\mathfrak X}$ (not necessarily symmetric), and $\Gamma$ denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under the assumption that $A$ belongs to ${\cal M}$, we describe the combinatorial structure of $\Gamma$. Moreover, we provide an algebraic-combinatorial characterization of $\Gamma$ when $A$ generates ${\cal M}$. Among else, we show that, if ${\mathfrak X}$ is a commutative $3$-class association scheme that is not an amorphic symmetric scheme, then we can always find a (directed) graph $\Gamma$ such that the adjacency matrix $A$ of $\Gamma$ generates the Bose--Mesner algebra ${\cal M}$ of ${\mathfrak X}$.

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Giusy Monzillo, Safet Penjić. 2023-07-21. On commutative association schemes and associated (directed) graphs. https://arxiv.org/abs/2307.11680

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