Work

On Commutative Association Schemes and Associated (Directed) Graphs

Year: 2025

Type: article

Abstract: Let $\mathcal{M}$ 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 $\mathcal{M}$, we describe the combinatorial structure of $\Gamma$. Moreover, we provide an algebraic-combinatorial characterization of $\Gamma$ when $A$ generates $\mathcal{M}$. 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 $\mathcal{M}$ of ${\mathfrak X}$.

Source: The Electronic Journal of Combinatorics

DOI: 10.37236/12973

Authors: Giusy Monzillo, Safet Penjić

Institution: University of Primorska

Language: English

Topic: Finite Group Theory Research

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Javna Agencija za Raziskovalno Dejavnost RS

Awards: Association Schemes, Automata and Incidnece Structures, Some applications of t-point counts in algebraic and combinatorial graph theory from the point of view of association schemes, Terwilliger algebra of a graph, Algebra, discrete mathematics, probability and game theory