On commutative association schemes and associated (directed) graphs

TL;DR

This paper explores the structure of graphs related to commutative association schemes, providing algebraic and combinatorial characterizations, especially focusing on cases where the adjacency matrix generates the Bose--Mesner algebra.

Contribution

It offers new insights into the structure of graphs associated with commutative association schemes, particularly when the adjacency matrix belongs to or generates the Bose--Mesner algebra.

Findings

If the scheme is a non-amorphic 3-class scheme, a graph exists with adjacency matrix generating the Bose--Mesner algebra.

Provides algebraic-combinatorial characterization of graphs when adjacency matrix belongs to the Bose--Mesner algebra.

Describes the structure of graphs when the adjacency matrix is in the Bose--Mesner algebra.

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 ${\mathfrak X}$.