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: 362; Downloads: 5
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2. On the uniform structure of bipartite graphs admitting a dual adjacency matrix candidateBlas Fernández, Roghayeh Maleki, Štefko Miklavič, Giusy Monzillo, 2026, original scientific article Abstract: Let Γ denote a finite, bipartite, connected graph with vertex set X. Fix x ∈ X and let ε ≥ 3 denote the eccentricity of x. For mutually distinct scalars {θ ∗ i }ε i=0 define a diagonal matrix A∗ = A∗(θ ∗ 0 , θ ∗ 1 , . . . , θ ∗ ε ) ∈ Mat X (R) as follows: for y ∈ X set (A∗)yy = θ ∗ ∂(x,y), where ∂ denotes the shortest path-length distance function of Γ. We say that A∗ is a dual adjacency matrix candidate of Γ with respect to x if the adjacency matrix A ∈ Mat X (R) of Γ and A∗ satisfy A3 A∗ − A∗ A3 + (β + 1)(A A∗ A2 − A2 A∗ A) = ρ(A A∗ − A∗ A) for some scalars β, ρ ∈ R. In this paper, we investigate when bipartite graphs that admit a dual adjacency matrix candidate also admit a uniform structure (in the sense of Terwilliger [6]). To do that, we first define a weakly uniform structure by slightly relaxing the conditions of uniform structure. The main result of this paper is that Γ admits a dual adjacency matrix candidate with respect to x if and only if Γ admits a weakly uniform structure with respect to x whose parameters satisfy some additional conditions. In particular, for β = 2, the weakly uniform structure is indeed a uniform structure.
Keywords: uniform property, dual adjacency matrix, Q-polynomial property
Published in RUP: 18.06.2026; Views: 489; Downloads: 10
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4. O nekaterih problemih, ki so povezani s Terwilligerjevimi algebrami in razdaljno-uravnoteženimi grafi : doktorska disertacijaBlas Fernández, 2023, doctoral dissertation Keywords: distance-regularized vertex, pseudo-distance-regularized vertex, Terwilliger algebra, irreducible module, distance-balanced graph, nicely distance-balanced graph, strongly distance-balanced graph
Published in RUP: 15.02.2023; Views: 4600; Downloads: 146
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