1. Every Q-polynomial distance-regular graph is sharp over $\mathbb{R}$Blas Fernández, Jae-Ho Lee, Jongyook Park, 2026, original scientific article Abstract: Let $\Gamma$ be a $Q$-polynomial distance-regular graph, and let $T=T(x)$ denote its Terwilliger algebra with respect to a fixed vertex $x$. While it has long been known that every irreducible $T$-module over the complex field is sharp, the corresponding result over the real field had remained unproved. In this work, we establish that every irreducible $T$-module over $\mathbb{R}$ is also sharp. This resolves the real analogue of a theorem of Nomura and Terwilliger and shows that every $Q$-polynomial distance-regular graph is sharp over both $\mathbb{R}$ and $\mathbb{C}$. As further consequences, we prove that the complexification of an irreducible real $T$-module remains irreducible, characterize isomorphism classes via complexification, determine the Wedderburn decomposition of the real Terwilliger algebra, and show that several naturally arising subalgebras are commutative and consist entirely of symmetric matrices. These results clarify the relationship between the real and complex representation theories of the Terwilliger algebra and
provide new structural insight into $Q$-polynomial distance-regular graphs.
Keywords: distance-regular graphs, Q-polynomial property, Terwilliger algebra
Published in RUP: 17.07.2026; Views: 14; Downloads: 1
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2. A classification of Q-polynomial distance-regular graphs with girth 6Štefko Miklavič, 2025, original scientific article Abstract: Let Γ denote a Q-polynomial distance-regular graph with diameter D and valency k≥3. In [Homotopy in Q-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189–206], H. Lewis showed that the girth of Γ is at most 6. In this paper we classify graphs that attain this upper bound. We show that Γ has girth 6 if and only if it is either isomorphic to the Odd graph on a set of cardinality 2D+1, or to a generalized hexagon of order (1,k−1).
Keywords: distance-regular graphs, Q-polynomial property, girth
Published in RUP: 01.12.2025; Views: 2456; Downloads: 3
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3. On commutative association schemes and associated (directed) graphsGiusy Monzillo, Safet Penjić, 2025, original scientific 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 ${\mathfrak X}$.
Keywords: commutative association schemes, association schemes, Bose-Mesner algebra, equitable partition, graphs generating schemes, quotient-polynomial graphs, x-distance-faithful intersection diagram
Published in RUP: 26.09.2025; Views: 805; Downloads: 4
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4. The distance function on Coxeter-like graphs and self-dual codesMarko Orel, Draženka Višnjić, 2025, original scientific article Keywords: Coxeter graph, invertible symmetric matrices, binary field, rank, distance in graphs, alternate matrices, self-dual codes
Published in RUP: 30.05.2025; Views: 1272; Downloads: 21
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7. O ekstremnih grafih z dano stopnjo in premerom/ožino : doktorska disertacijaSlobodan Filipovski, 2018, doctoral dissertation Keywords: adjacency matrix, antipodal graphs, cages, excess, defect, Ramanujan graphs, selfrepeats, degree/diameter problem, spectrum, Moore graphs, asymptotic density, distance matrices, Bermond and Bollobas problem
Published in RUP: 21.01.2019; Views: 6112; Downloads: 0 |
8. On the Terwilliger algebra of bipartite distance-regular graphs with $G_{i-1,i-1,1}(x, y, z) = alpha_i + beta_i G_{1,1,i%1}(x, y, z)$ : 31th Ljubljana - Leoben Graph Theory Seminar, September 3-4, 2018, LoebenSafet Penjić, 2018, unpublished conference contribution Keywords: distance-regular graphs, Terwilliger algebra, irreducible modules
Published in RUP: 17.09.2018; Views: 3500; Downloads: 32
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10. On the Terwilliger algebra of bipartite distance-regular graphs with [Delta][sub]2 = 0 and c[sub]2=1Mark MacLean, Štefko Miklavič, Safet Penjić, 2016, original scientific article Abstract: Let ▫$\Gamma$▫ denote a bipartite distance-regular graph with diameter ▫$D \geq 4$▫ and valency ▫$k \geq 3$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫, and let ▫$A$▫ denote the adjacency matrix of ▫$\Gamma$▫. For ▫$x \in X$▫ and for ▫$0 \leq i \leq D$▫, let ▫$\operatorname{\Gamma}_i(x)$▫ denote the set of vertices in ▫$X$▫ that are distance ▫$i$▫ from vertex ▫$x$▫. Define a parameter ▫$\operatorname{\Delta}_2$▫ in terms of the intersection numbers by ▫$\operatorname{\Delta}_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$▫. We first show that ▫$\operatorname{\Delta}_2 = 0$▫ implies that ▫$D \leq 5$▫ or ▫$c_2 \in \{1, 2 \}$▫. For ▫$x \in X$▫ let ▫$T = T(x)$▫ denote the subalgebra of ▫$\text{Mat}_X(\mathbb{C})$▫ generated by ▫$A, E_0^\ast, E_1^\ast, \ldots, E_D^\ast$▫, where for ▫$0 \leq i \leq D$, $E_i^\ast$▫ represents the projection onto the▫ $i$▫th subconstituent of ▫$\Gamma$▫ with respect to ▫$x$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. By the endpoint of an irreducible ▫$T$▫-module ▫$W$▫ we mean ▫$\min \{i | E_i^\ast W \ne 0 \}$▫. In this paper we assume ▫$\Gamma$▫ has the property that for ▫$2 \leq i \leq D - 1$▫, there exist complex scalars ▫$\alpha_i$▫, ▫$\beta_i$▫ such that for all ▫$x, y, z \in X$▫ with ▫$\partial(x, y) = 2$▫, ▫$\partial(x, z) = i$▫, ▫$\partial(y, z) = i$▫, we have ▫$\alpha_i + \beta_i | \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i - 1}(z) | = | \operatorname{\Gamma}_{i - 1}(x) \cap \operatorname{\Gamma}_{i - 1}(y) \cap \operatorname{\Gamma}_1(z) |$▫. We additionally assume that▫ $\operatorname{\Delta}_2 = 0$▫ with ▫$c_2 = 1$▫. Under the above assumptions we study the algebra ▫$T$▫. We show that if ▫$\Gamma$▫ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible ▫$T$▫-module with endpoint 2. We give an orthogonal basis for this ▫$T$▫-module, and we give the action of ▫$A$▫ on this basis.
Keywords: distance-regular graphs, terwilliger algebra, subconstituent algebra
Published in RUP: 14.11.2017; Views: 4471; Downloads: 150
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