1. On cubic polycirculant nut graphsNino Bašić, Ivan Damnjanović, 2025, original scientific article Abstract: A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms having $\ell$ vertex orbits of equal size. It is not difficult to observe that there exists no cubic $1$-circulant nut graph or cubic $2$-circulant nut graph, while the full classification of all the cubic $3$-circulant nut graphs was recently obtained (Damnjanović et al. in Electron. J. Comb. 31(2):P2.31, 2024). Here, we investigate the existence of cubic $\ell$-circulant nut graphs for $\ell \geq 4$ and show that there is no cubic $4$-circulant nut graph or cubic $5$-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic $\ell$-circulant nut graphs for any fixed $\ell \in \{6, 7\}$ or $\ell \geq 9$.
Keywords: nut graph, polycirculant graph, cubic graph, pregraph, voltage graph
Published in RUP: 19.11.2025; Views: 265; Downloads: 5
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2. Polycyclic geometric realizations of the Gray configurationLeah Berman, Gábor Gévay, Tomaž Pisanski, 2025, original scientific article Abstract: The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3×3×3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z3 and Z9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z9 symmetry is only a “weak” realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration).
Keywords: Gray graph, Gray configuration, polycirculant, polycyclic configuration
Published in RUP: 29.09.2025; Views: 426; Downloads: 4
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3. The Gray graph is a unit-distance graphLeah Berman, Gábor Gévay, Tomaž Pisanski, 2025, original scientific article Abstract: In this note we give a construction proving that the Gray graph, which is the smallestcubic semisymmetric graph, is a unit-distance graph.
Keywords: polycirculant, unit-distance graph, Gray graph, ADAM graph, generalized Petersen graph
Published in RUP: 10.09.2025; Views: 515; Downloads: 2
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