1. Nut graphs with a prescribed number of vertex and edge orbitsNino Bašić, Ivan Damnjanović, 2026, original scientific article Abstract: A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even $r \geq 2$ and any $k \geq r + 1$, there exist infinitely many nut graphs with r vertex orbits and k edge orbits. Here, we extend this result by finding all the pairs $(r, k)$ for which there exists a nut graph with $r$ vertex orbits and $k$ edge orbits. In particular, we show that for any $k \geq 2$, there are infinitely many Cayley nut graphs with $k$ edge orbits and $k$ arc orbits.
Keywords: nut graph, vertex orbit, edge orbit, arc orbit, Cayley graph, automorphism
Published in RUP: 09.01.2026; Views: 290; Downloads: 5
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2. 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: 543; Downloads: 4
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3. Regular maps with primitive automorphism groupsGareth A. Jones, Martin Mačaj, 2025, original scientific article Abstract: We classify the regular maps ℳ which have automorphism groups G acting faithfully and primitively on their vertices. As a permutation group G must be of almost simple or affine type, with dihedral point stabilisers. We show that all such almost simple groups, namely all but a few groups PSL2(q), PGL2(q) and Sz(q), arise from regular maps, which are always non-orientable. In the affine case, the maps ℳ occur in orientable and non-orientable Petrie dual pairs. We give the number of maps associated with each group, together with their genus and extended type. Some of this builds on earlier work of the first author on generalised Paley maps, and on recent work of Jajcay, Li, Širáň and Wang on maps with quasiprimitive automorphism groups.
Keywords: regular map, automorphism group, primitive, almost simple, affine group
Published in RUP: 04.11.2025; Views: 405; Downloads: 2
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4. Colour-permuting automorphisms of complete Cayley graphsShirin Alimirzaei, Dave Witte Morris, 2025, original scientific article Abstract: Let G be a (finite or infinite) group, and let KG = Cay(G; G \ {1}) be the complete graph with vertex set G, considered as a Cayley graph of G. Being a Cayley graph, it has a natural edge-colouring by sets of the form {s, s-1} for s in G. We prove that every colour-permuting automorphism of KG is an affine map, unless G is isomoprhic to the direct product of Q8 and B, where Q8 is the quaternion group of order 8, and B is an abelian group, such that b2 is trivial for all b in B.
We also prove (without any restriction on G) that every colour-permuting automorphism of KG is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D. P. Byrne, M. J. Donner, and T. Q. Sibley in 2013.
Keywords: Cayley graph, automorphism, colour-permuting, complete graphs
Published in RUP: 03.11.2025; Views: 326; 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: 398; Downloads: 1
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6. The invariant of PGU(3,q) in the Hermitian function fieldBarbara Gatti, Francesco Ghiandoni, Gábor Korchmáros, 2025, original scientific article Abstract: Let F = F|K be a function field over an algebraically closed constant field K of positive characteristic p. For a K-automorphism group G of F, the invariant of G is the fixed field FG of G. If F has transendency degree 1 (i.e. F is the function field of an irreducible curve) and FG is rational, then each generator of FG uniquely determines FG and it makes sense to call each of them the invariant of G. In this paper, F is the Hermitian function field K(Hq)=K(x,y) with yq + y − xq + 1 = 0 and q = pr. We determine the invariant of Aut(K(Hq)) \cong PGU(3,q), and discuss some related questions on Galois subcovers of maximal curves over finite fields.
Keywords: function field, finite field, automorphism group, invariant
Published in RUP: 21.10.2025; Views: 335; Downloads: 4
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9. 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: 1999; Downloads: 35
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