1. Scattered polynomials: an overview on their properties, connections and applicationsGiovanni Longobardi, 2026, original scientific article Abstract: The aim of this survey is to outline the state of the art in research on a class of linearized polynomials with coefficients over finite fields, known as scattered polynomials. These have been studied in several contexts, such as those described by A. Blokhuis, M. Lavrauw in 2000, and by G. Lunardon, O. Polverino in 2001. In 2016, J. Sheekey brought to light their connection to maximum rank-metric codes. This link has significantly advanced their study and investigation, sparking considerable interest in recent years.
Here, we will explore their relationship with certain subsets of the finite projective line PG(1,qn) known as maximum scattered linear sets, as well as with codes made up of square matrices of order n equipped with the rank metric. We will review the known examples of scattered polynomials known to date and discuss some of their key properties. We will also address the classification of maximum scattered linear sets of the finite projective line PG(1,qn) for small values of n and discuss characterization results for the examples known so far.
Finally, we will retrace how each scattered polynomial gives rise to a translation plane, as discussed by V. Casarino, G. Longobardi, C. Zanella in 2022 and by G. Longobardi, C. Zanella in 2024.
Keywords: finite field, linearized polynomial, scattered polynomial, projective space, linear set, rank-metric code
Published in RUP: 16.09.2026; Views: 335; Downloads: 9
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2. 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: 485; Downloads: 5
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3. 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: 600; Downloads: 12
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5. 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: 2805; Downloads: 5
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7. The independence polynomial of trees is not always log-concave starting from order 26Ohr Kadrawi, Vadim Levit, 2025, original scientific article Abstract: An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in G is represented by α(G). The independence polynomial of a graph G = (V, E) was introduced by Gutman and Harary in 1983 and is defined as
I(G; x) = Σ_{k = 0}^α(G) s_k x^k = s₀ + s₁x + s₂x² + ... + s_α(G)x^α(G), where sk represents the number of independent sets in G of size k.
The problem raised by Alavi, Malde, Schwenk, and Erdös in 1987 stated that the independence polynomials of trees are unimodal, and many researchers believed that this problem could be strengthened up to its corresponding log-concave version. However, in 2023, this conjecture was shown to be false by Kadrawi, Levit, Yosef, and Mizrachi. In this paper, we provide further evidence against this conjecture by presenting infinite families of trees with independence polynomials that are not log-concave.
Keywords: tree, independent set, independence polynomial, unimodality, log-concavity
Published in RUP: 22.10.2025; Views: 1923; Downloads: 10
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8. 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: 1102; Downloads: 7
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9. On ▫$L^2$▫ approximation by planar Pythagorean-hodograph curvesRida T. Farouki, Marjetka Knez, Vito Vitrih, Emil Žagar, 2025, original scientific article Abstract: The ▫$L^2$▫ approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex valued functions of a real parameter. Because of the nonlinear nature of polynomial PH curves, constructing ▫$L^2$▫ approximants involves solving a nonlinear optimization problem. However, a simplified method that requires only the solution of a linear system may be developed by formulating the ▫$L^2$▫ approximation in the preimage space. The extension of the methodology to approximation by PH B-spline curves is also addressed, and several examples are provided to illustrate its implementation and potential.
Keywords: ▫$L^2$▫ approximation, complex polynomial, Pythagorean-hodograph curve, Pythagorean-hodograph spline, preimage
Published in RUP: 30.05.2025; Views: 3252; Downloads: 36
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10. A complexitystudy of distance variants of covering and domination problems in H-free graphs : master's thesisMirza Krbezlija, 2021, master's thesis Keywords: distance-k dominating set, distance-k edge dominating set, distance-k vertex cover, distance k-edge cover, H-free graph, polynomial-time algorithm, NP-complete problem, dichotomy theorem
Published in RUP: 18.10.2021; Views: 6162; Downloads: 66
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