1. A skew group ring of Z/2Z over U(sl_2), Leonard triples and odd graphsHau-Wen Huang, Chin-Yen Lee, 2026, izvirni znanstveni članek Opis: We employ a skew group ring of Z/2Z over U(sl₂) to construct modules over the universal Bannai–Ito algebra. Furthermore, we characterize the conditions under which the defining generators act as Leonard triples on these modules. As a combinatorial realization, we establish a surjective algebra homomorphism from the universal Bannai–Ito algebra to the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.
Ključne besede: Leonard triples, odd graphs, skew group rings, Terwilliger algebras
Objavljeno v RUP: 28.09.2026; Ogledov: 146; Prenosov: 2
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2. Connect four on treesRobert A. Beeler, Rodney Keaton, 2026, izvirni znanstveni članek Opis: Connect Four is a two player table game. Players alternate dropping checkers into a 6 X 7 vertical grid. The first player to have four of their respective pieces on a horizontal, vertical, or diagonal row wins the game. In this paper, we consider playing Connect Four on trees. We begin by determining the outcome of the game when played on trees of diameter five or less. We are also able to classify all trees on which the first player can win in their first four moves. Finally, we give a construction of an infinite family of trees on which the first player can win.
Ključne besede: combinatorial games, games on graphs, connect four.
Objavljeno v RUP: 23.09.2026; Ogledov: 219; Prenosov: 6
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3. Tree-independence number of ▫$P_5$▫-free graphs with no large bicliquesVáclav Blažej, J. Pascal Gollin, Tomáš Hons, Tomáš Masařík, Martin Milanič, Paweł Rzążewski, Ondřej Suchý, Alexandra Wesolek, 2026, objavljeni znanstveni prispevek na konferenci Opis: The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique K_{t,t} forces tree-independence number at least t. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers s and t, {P_s,K_{t,t}}-free graphs have bounded tree-independence number. We prove this conjecture for s = 5 by showing that every {P₅,K_{t,t}}-free graph has tree-independence number at most 4t. We also obtain related bounds for the weaker parameter of independence degeneracy.
Ključne besede: tree-independence number, independence degeneracy, independence treewidth, P5-free graphs
Objavljeno v RUP: 21.09.2026; Ogledov: 274; Prenosov: 5
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4. Edge criticality in signed graphs admitting a Roman dominating functionJames Joseph, Mayamma Joseph, 2026, izvirni znanstveni članek Opis: A Roman dominating function (RDF) on a signed graph S = (G, σ) is a function f: V(S) → {0, 1, 2} such that f(N[v]) ≥ 1 for every vertex v ∈ V(S) and any vertex v with f(v) = 0 has a neighbour u ∈ N⁺(v) having f(u) = 2, where f(N[v]) = f(v) + ∑u ∈ N(v) σ(uv)f(u). The weight of an RDF is ω(f) = ∑v ∈ V f(v) and the minimum weight among all the RDFs on S is called the Roman domination number, γ_R(S). In this article we explore the concept of edge criticality in signed graphs admitting an RDF by examining the signed graphs S such that γ_R(S + uv) < γ_R(S), for any pair of non-adjacent vertices u and v of S, such that the edge uv is positive.
Ključne besede: signed graphs, roman dominating function, roman domination number, roman domination edge critical graphs
Objavljeno v RUP: 16.09.2026; Ogledov: 229; Prenosov: 7
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5. Domination in cylindrical graphsJosé Antonio Martínez, Mercè Mora, María Luz Puertas, Javier Tejel, 2026, izvirni znanstveni članek Opis: The domination number γ(Cm □ Pn) of the Cartesian product Cm □ Pn of a cycle and a path has been computed when m ≡ 0, 2 (mod 5). In the remaining cases m ≡ 1, 3, 4 (mod 5), exact formulae for γ(Cm □ Pn) have been determined when either n ≤ 22 or m ≤ 30. For the rest of the cases, only lower and upper bounds for γ(Cm □ Pn) are known. In this paper, we study γ(Cm □ Pn) when m ≡ 1, 3, 4 (mod 5). In particular, we compute γ(Cm □ Pn) if 30 ≤ m ≡ 1 (mod 5) and n ≥ 22, and we provide tighter lower and upper bounds for γ(Cm □ Pn) if m ≡ 3, 4 (mod 5).
Ključne besede: Domination in graphs, Cartesian product graphs, tropical matrix multiplication
Objavljeno v RUP: 20.08.2026; Ogledov: 390; Prenosov: 10
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6. Extending graph burning to hypergraphsAndrea C. Burgess, Caleb W. Jones, David A. Pike, 2026, izvirni znanstveni članek Opis: Graph burning is a round-based game or process that discretely models the spread of influence throughout a network. We introduce a generalization of graph burning which applies to hypergraphs, as well as a variant called "lazy" hypergraph burning. Interestingly, lazily burning a graph is trivial, while lazily burning a hypergraph can be quite complicated. Moreover, the lazy burning model is a useful tool for analyzing the round-based model. One of our key results is that arbitrary hypergraphs do not satisfy a bound analogous to the one in the Burning Number Conjecture for graphs. We also obtain bounds on the burning number and lazy burning number of a hypergraph in terms of its parameters, and present several open problems in the field of (lazy) hypergraph burning.
Ključne besede: Combinatorial games on graphs, pursuit-evasion, graph searching, graph burning, hypergraph theory
Objavljeno v RUP: 18.08.2026; Ogledov: 464; Prenosov: 8
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7. Some results on ▫$\sigma_t$▫-irregularitySlobodan Filipovski, Darko Dimitrov, Martin Knor, Riste Škrekovski, 2026, izvirni znanstveni članek Opis: The (\sigma_t)-irregularity (or sigma total index) is a graph invariant defined as [ \sigma_t(G)=\sum_{{u,v}\subseteq V(G)}(d(u)-d(v))^2, ] where (d(z)) denotes the degree of a vertex (z). This irregularity measure was proposed by Réti in 2019 and recently rediscovered by Dimitrov and Stevanović in 2023. In this paper, we remark that (\sigma_t(G)=n^2\operatorname{Var}(G)), where (\operatorname{Var}(G)) is the degree variance of the graph. We show that among all complete bipartite graphs on (n) vertices, one of the corresponding complete bipartite graphs whose part sizes are closest to (n(2-\sqrt{2})/4) and (n(2+\sqrt{2})/4) has the maximum sigma total index. Moreover, various upper and lower bounds for (\sigma_t)-irregularity are provided. In this direction, we establish a relation between the graph energy (\mathcal{E}(G)) and (\sigma_t)-irregularity and derive bounds related to the Laplacian eigenvalues of the graph.
Ključne besede: irregularity, total irregularity, energy of graphs, Laplacian eigenvalues
Objavljeno v RUP: 14.08.2026; Ogledov: 370; Prenosov: 6
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8. Every Q-polynomial distance-regular graph is sharp over $\mathbb{R}$Blas Fernández, Jae-Ho Lee, Jongyook Park, 2026, izvirni znanstveni članek Opis: 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.
Ključne besede: distance-regular graphs, Q-polynomial property, Terwilliger algebra
Objavljeno v RUP: 17.07.2026; Ogledov: 447; Prenosov: 5
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9. Uniform equations for bipartite graphs and the center of a Terwilliger algebraŠtefko Miklavič, Giusy Monzillo, 2026, izvirni znanstveni članek Opis: The uniform property was introduced by P. Terwilliger in the context of graded posets and was later extended to connected bipartite graphs. The core of this definition involves the so called uniform equations that must be satisfied. Let Γ denote a connected bipartite graph. Fix a vertex x of Γand let T=T(x) denote the corresponding Terwilliger algebra. In this paper, we study the connections between the uniform equations and the center of T. We show that these uniform equations give rise to a certain subspace of the center of T. Changing the logical direction, we show that if a matrix of a particular form belongs to the center of T, then uniform equations are satisfified.
Ključne besede: uniform equations, center of a Terwilliger algebra, bipartite graphs
Objavljeno v RUP: 08.05.2026; Ogledov: 584; Prenosov: 19
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