1. Nut graphs with a given automorphism groupNino Bašić, Patrick W. Fowler, 2025, original scientific article Abstract: A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all nonzero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group can be represented as the group of automorphisms of infinitely many nut graphs. It is further shown that such nut graphs exist even within the class of regular graphs; the cases where the degree is 8, 12, 16, 20 or 24 are realised explicitly.
Keywords: nut graph, graph automorphism, automorphism group, nullity, graph spectra, f-universal
Published in RUP: 25.11.2025; Views: 426; Downloads: 4
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2. A unified Erdős–Pósa theorem for cycles in graphs labelled by multiple abelian groupsJ. Pascal Gollin, Kevin Hendrey, O-joung Kwon, Sang-il Oum, Youngho Yoo, 2025, original scientific article Abstract: In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs (l, z) of integers where such a duality holds for the family of cycles of length l modulo z. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.
Keywords: Erdős-Pósa property, cycle packing, group-labelled graph
Published in RUP: 17.11.2025; Views: 337; Downloads: 8
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3. Transitive regular q-analogs of graphsDean Crnković, Vedrana Mikulić Crnković, Andrea Švob, Matea Zubović Žutolija, 2025, original scientific article Abstract: In 1976, Delsarte introduced the notion of q-analogs of designs, and q-analogs of graphs were introduced recently by M. Braun et al. In this paper, we extend that study by giving a method for constructing transitive regular q-analogs of graphs. Further, we illustrate the method by giving some examples. Additionally, we introduced the notion of q-analogs of quasi-strongly regular graphs and give examples of transitive q-analogs of quasi-strongly regular graphs coming from spreads.
Keywords: q-ary design, q-ary graph, regular graph, transitive group
Published in RUP: 03.11.2025; Views: 225; Downloads: 1
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4. Minimal directed strongly regular Cayley graphs over generalized dicyclic groupsYueli Han, Lu Lu, 2025, original scientific article Abstract: Let G be a group with identity element 1, and let S be a subset of G \ {1}. The subset S is called minimal if ⟨S⟩ = G and there exists an element s ∈ S such that ⟨S \ {s, s−1}⟩ ≠ G. In this paper, we completely determine all directed strongly regular Cayley graphs Cay(G, S) for any generalized dicyclic group G, provided that S is a minimal subset of G.
Keywords: directed strongly regular graph, Cayley graph, generalized dicyclic group
Published in RUP: 21.10.2025; Views: 329; Downloads: 2
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5. Regular and semi-regular representations of groups by posetsJonathan A. Barmak, 2025, original scientific article Abstract: By a result of Babai, with finitely many exceptions, every group G admits a semi-regular poset representation with three orbits, that is, a poset P with automorphism group Aut(P) ≃ G such that the action of Aut(P) on the underlying set is free and with three orbits. Among finite groups, only the trivial group and ℤ_2 have a regular poset representation (i.e. semi-regular with one orbit), however many infinite groups admit such a representation. In this paper we study non-necessarily finite groups which have a regular representation or a semi-regular representation with two orbits. We prove that if G admits a Cayley graph which is locally the Cayley graph of a free group, then it has a semi-regular representation of height 1 with two orbits. In this case we will see that any extension of the integers by G admits a regular representation. Applications are given to finite simple groups, hyperbolic groups, random groups and indicable groups.
Keywords: automorphism group of posets, Cayley graph, Dehn presentation, simple groups, random groups
Published in RUP: 21.10.2025; Views: 307; Downloads: 0
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9. Cyclic m-DCI-groups and m-CI-groupsIstván Kovács, Luka Šinkovec, 2025, original scientific article Abstract: Based on the earlier work of Li from 1997 and Dobson from 2008, in this paper we complete the classification of cyclic m-DCI-groups and m-CI-groups. For a positive integer m such that m ≥ 3, we show that the group ℤ_(n) is an m-DCI-group if and only if n is not divisible by 8 nor by p² for any odd prime p < m. Furthermore, if m ≥ 6, then we show that ℤn is an m-CI-group if and only if either n ∈ {8, 9, 18}, or n ∉ {8, 9, 18} and n is not divisible by 8 nor by p² for any odd prime p < (m - 1)/2.
Keywords: Cayley graph, cyclic group, m-CI-group, m-DCI-group
Published in RUP: 01.04.2025; Views: 1064; Downloads: 12
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10. Partial geometries with regular automorphism groups : master’s thesisAdisa Hodžić, 2024, master's thesis Keywords: (near-) linear space, projective plane, affine plane, partial geometry, generalized quadrangle, strongly regular graph, partial difference set, automorphism group
Published in RUP: 25.12.2024; Views: 1872; Downloads: 31
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