1. Uniform equations for bipartite graphs and the center of a Terwilliger algebraŠtefko Miklavič, Giusy Monzillo, 2026, original scientific article Abstract: 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.
Keywords: uniform equations, center of a Terwilliger algebra, bipartite graphs
Published in RUP: 08.05.2026; Views: 263; Downloads: 8
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2. Group distance magic cubic graphsSylwia Cichacz, Štefko Miklavič, 2026, original scientific article Abstract: A $\Gamma$-distance magic labeling of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$, for which there exists $\mu \in \Gamma$, such that the weight $w(x) =\sum_{y\in N(x)}\ell(y)$ of every vertex $x \in V$ is equal to $\mu$. In this case, the element $\mu$ is called the magic constant of $G$. A graph $G$ is called a group distance magic if there exists a $\Gamma$-distance magic labeling of $G$ for every Abelian group $\Gamma$ of order $n$. In this paper, we focused on cubic $\Gamma$-distance magic graphs as well as some properties of such graphs.
Keywords: group distance magic labeling, Kotzig array, generalized Petersen graph
Published in RUP: 06.05.2026; Views: 306; Downloads: 6
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4. 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: 2302; Downloads: 3
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5. On 3-isoregularity of multicirculantsKlavdija Kutnar, Dragan Marušič, Štefko Miklavič, 2025, original scientific article Abstract: A graph is said to be k-isoregular if any two vertex subsets of cardinality at most k, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no 3-isoregular bicirculant (and more generally, no locally 3-isoregular bicirculant) of order twice an odd number exists. Further, partial results for bicirculants of order twice an even number as well as tricirculants of specific orders, are also obtained. Since 3-isoregular graphs are necessarily strongly regular, a motivation for the above result about bicirculants is that it brings us a step closer to obtaining a direct proof of a classical consequence of the Classification of Finite Simple Groups, that no simply primitive group of degree twice a prime exists for primes greater than 5.
Keywords: 3-isoregularity, strongly regular graph, bicirculant, tricirculant
Published in RUP: 06.08.2025; Views: 776; Downloads: 6
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