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1.
Classification of cubic vertex-transitive tricirculants
Primož Potočnik, Micael Toledo, 2020, original scientific article

Keywords: graph, cubic, semiregular automorphism, tricirculant, vertex-transitive
Published in RUP: 03.01.2022; Views: 795; Downloads: 40
.pdf Full text (1,18 MB)

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Some recent discoveries about half-arc-transitive graphs : dedicated to Dragan Marušič on the occasion of his 60th birthday
Marston D. E. Conder, Primož Potočnik, Primož Šparl, 2015, original scientific article

Abstract: We present some new discoveries about graphs that are half-arc-transitive (that is, vertex- and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser ▫$D_4$▫ (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser of order 16 that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.
Keywords: graph, edge-transitive, vertex-transitive, arc-transitive, half arc-transitive
Published in RUP: 31.12.2021; Views: 743; Downloads: 16
.pdf Full text (333,06 KB)

4.
A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two : dedicated to Dragan Marušič on the occasion of his 60th birthday
Primož Potočnik, Pablo Spiga, Gabriel Verret, 2015, original scientific article

Abstract: A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a half-arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed.
Keywords: graphs, digraphs, edge-transitive, vertex-transitive, arc-transitive, half arc-transitive
Published in RUP: 31.12.2021; Views: 792; Downloads: 16
.pdf Full text (370,47 KB)

5.
Reachability relations, transitive digraphs and groups
Aleksander Malnič, Primož Potočnik, Norbert Seifter, Primož Šparl, 2015, original scientific article

Abstract: In [A. Malnič, D. Marušič, N. Seifter, P. Šparl and B. Zgrablič, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566-1581] it was shown that properties of digraphs such as growth, property ▫$\mathbf{Z}$▫, and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs. In this paper we study these relations in connection with certain properties of automorphism groups of transitive digraphs. In particular, one of the main results shows that if atransitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, then its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question. The obtained results have interesting implications for Cayley digraphs of certain types of groups such as torsion-free groups of polynomial growth.
Keywords: Cayley digraph, reachability relation
Published in RUP: 31.12.2021; Views: 772; Downloads: 16
.pdf Full text (311,92 KB)

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Arc-transitive digraphs of given out-valency and with blocks of given size
Luke Morgan, Primož Potočnik, Gabriel Verret, 2019, original scientific article

Abstract: Given integers ▫$k$▫ and ▫$m$▫, we construct a ▫$G$▫-arc-transitive graph of valency ▫$k$▫ and an ▫$L$▫-arc-transitive oriented digraph of out-valency ▫$k$▫ such that ▫$G$▫ and ▫$L$▫ both admit blocks of imprimitivity of size ▫$m$▫.
Keywords: arc-transitive digraphs, Cayley digraphs, imprimitive digraphs
Published in RUP: 28.06.2019; Views: 1557; Downloads: 126
URL Link to full text

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On reflexible polynomials
Aleksander Malnič, Boštjan Kuzman, Primož Potočnik, 2018, published scientific conference contribution abstract (invited lecture)

Keywords: minimal elementary abelian cover, doubled cycle, polynomial
Published in RUP: 07.02.2018; Views: 2886; Downloads: 30
URL Link to full text

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