1. Classification of pentavalent symmetric tricirculantsYasamin Khaefi, Klavdija Kutnar, Dragan Marušič, 2026, original scientific article Abstract: A graph $\Gamma$ is said to be an {\em $m$-Cayley graph} on a group $G$ ($|G|\ne 1$) if its automorphism group contains a semiregular subgroup isomorphic to $G$ having $m$ orbits on the vertex set of $\Gamma$. If $G$ is cyclic and $m=3$ then $\Gamma$ is called a {\em tricirculant}. A graph is said to be {\em symmetric} if its automorphism group acts transitively on the set of its arcs. In this paper, it is shown that with the exception of $K_6$, no connected pentavalent symmetric tricirculant exists.
Keywords: pentavalent graph, symmetric, semiregular automorphism, tricirculant
Published in RUP: 22.06.2026; Views: 483; Downloads: 10
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2. Platonic configurations of points and linesJurij Kovič, Aleksander Simonič, 2026, original scientific article Abstract: We present some methods for constructing connected spatial geometric configurations (p_q, n_k) of points and lines, preserved by the same isometries of Euclidean space E³ as the predetermined Platonic solid. In this paper, we are mainly interested in configurations (n₃), (n₄), and (n₅), but also in unbalanced configurations (p₃, n₄), (p₃, n₅), and (p₄, n₅).
Keywords: configuration of points and lines, symmetry group, Platonic solid, centrally symmetric solid, projection from a point
Published in RUP: 22.12.2025; Views: 721; Downloads: 5
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3. Bounding s for vertex-primitive s-arc-transitive digraphs of alternating and symmetric groupsJunyan Chen, Lei Chen, Michael Giudici, Jing Jian Li, Cheryl E. Praeger, Binzhou Xia, 2025, original scientific article Abstract: Determining an upper bound on s for finite vertex-primitive s-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on s is attained for some digraph admitting an almost simple s-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that s<=2 in the case where the group is an alternating or symmetric group.
Keywords: digraph, vertex-primitive, s-arc-transitive, alternating group, symmetric group
Published in RUP: 22.10.2025; Views: 888; Downloads: 9
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4. Cubes of symmetric designsVedran Krčadinac, Mario Osvin Pavčević, Kristijan Tabak, 2025, original scientific article Abstract: We study n-dimensional matrices with {0, 1}-entries (n-cubes) such that all their 2-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting n-cubes may have inequivalent slices. For suitable parameters, they can be transformed into n-dimensional Hadamard matrices with this property. In contrast, previously known constructions of n-dimensional designs all give examples with equivalent slices.
Keywords: symmetric design, difference set, Hadamard matrix
Published in RUP: 21.10.2025; Views: 662; Downloads: 7
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5. Homomorphisms from the Coxeter graphMarko Orel, Draženka Višnjić, 2025, original scientific article Abstract: Let $S_n(\mathbb{F}_2)$ be the set of all $n\times n$ symmetric matrices with coefficients in the binary field $\mathbb{F}_2=\{0,1\}$, and let $SGL_n(\mathbb{F}_2)$ be its subset formed by invertible matrices. Let $\widehat{\Gamma}_n$ be the graph with the vertex set $S_n(\mathbb{F}_2)$ where a pair of vertices $\{A,B\}$ form an edge if and only if $rank(A-B)=1$. Similarly, let $\Gamma_n$ be the subgraph in $\widehat{\Gamma}_n$, which is induced by the set $SGL_n(\mathbb{F}_2)$. Graph $\Gamma_n$ generalizes the well-known Coxeter graph, which is isomorphic to $\Gamma_3$. Motivated by research topics in coding theory, matrix theory, and graph theory, this paper represents the first step towards the characterization of all graph homomorphisms $\Phi: \Gamma_n\to \widehat{\Gamma}_m$ where $n,m$ are positive integers. Here, the case $n=3$ is solved.
Keywords: preserver problems, symmetric matrices, invertible matrices, binary field, rank, graph homomorphisms, Coxeter graph
Published in RUP: 27.08.2025; Views: 1066; Downloads: 7
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7. 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: 1470; Downloads: 23
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8. Linking rings structures and semisymmetric graphs : combinatorial constructionsPrimož Potočnik, Steve Wilson, 2018, original scientific article Keywords: graphs, automorphism group, symmetry, locally arc-transitive graphs, symmetric graphs, cycle structure, linking ring structure
Published in RUP: 03.01.2022; Views: 3144; Downloads: 27
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