20.500.12556/RUP-21794 On commutative association schemes and associated (directed) graphs 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 ${\mathfrak X}$. commutative association schemes association schemes Bose-Mesner algebra equitable partition graphs generating schemes quotient-polynomial graphs x-distance-faithful intersection diagram komutativne asociativne sheme asociativne sheme Bose-Mesnerjeva algebra ekvitabilna particija kvocientno-polinomski grafi x-razdaljno-zvest diagram presekov true false false Angleški jezik Slovenski jezik Članek v reviji 2025-09-26 11:03:16 2025-09-26 11:03:18 2025-10-17 03:02:52 0000-00-00 00:00:00 2025 0 0 str. 1-38 iss. 1, article no. P1.54 Vol. 32 2025 0000-00-00 Zaloznikova NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2025-03-28 519.17 1077-8926 10.37236/12973 231013635 RAZ_Monzillo_Giusy_2025.pdf RAZ_Monzillo_Giusy_2025.pdf 1 2E879042E1B1DAEC669B54F3D013AA74 eb54f46d1d3a6d59e43e45cf682286e6b31fcce034012d6fe4fa7b32a4e3643a f887dd3e-9ab7-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31686 https://www.combinatorics.org/ojs/index.php/eljc/article/view/v32i1p54 1 c8319bb6-9ab7-11f0-8f0b-005056ac49c0 https://repozitorij.upr.si/Dokument.php?lang=slv&id=31684 Fakulteta za matematiko, naravoslovje in informacijske tehnologije Inštitut Andrej Marušič 0 0 0