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 ${\mathfrak X}$.
@article{10_37236_12973,
author = {Giusy Monzillo and Safet Penji\'c},
title = {On commutative association schemes and associated (directed) graphs},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {1},
doi = {10.37236/12973},
zbl = {1564.05371},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12973/}
}
TY - JOUR
AU - Giusy Monzillo
AU - Safet Penjić
TI - On commutative association schemes and associated (directed) graphs
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/12973/
DO - 10.37236/12973
ID - 10_37236_12973
ER -
%0 Journal Article
%A Giusy Monzillo
%A Safet Penjić
%T On commutative association schemes and associated (directed) graphs
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/12973/
%R 10.37236/12973
%F 10_37236_12973
Giusy Monzillo; Safet Penjić. On commutative association schemes and associated (directed) graphs. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/12973
