1. 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: 02.01.2022; Views: 1140; Downloads: 19 Full text (397,55 KB) |
2. Vertex-transitive graphs and their arc-typesMarston D. E. Conder, Tomaž Pisanski, Arjana Žitnik, 2017, original scientific article Abstract: Let ▫$X$▫ be a finite vertex-transitive graph of valency ▫$d$▫, and let ▫$A$▫ be the full automorphism group of ▫$X$▫. Then the arc-type of ▫$X$▫ is defined in terms of the sizes of the orbits of the stabiliser ▫$A_v$▫ of a given vertex ▫$v$▫ on the set of arcs incident with ▫$v$▫. Such an orbit is said to be self-paired if it is contained in an orbit ▫$\Delta$▫ of ▫$A$▫ on the set of all arcs of v$X$▫ such that v$\Delta$▫ is closed under arc-reversal. The arc-type of ▫$X$▫ is then the partition of ▫$d$▫ as the sum ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, where ▫$n_1, n_2, \dots, n_t$▫ are the sizes of the self-paired orbits, and ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ▫$1+1$▫ and ▫$(1+1)$▫, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type. Keywords: symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, cartesian product, covering graph Published in RUP: 02.01.2022; Views: 1167; Downloads: 20 Full text (475,17 KB) |
3. Some recent discoveries about half-arc-transitive graphs : dedicated to Dragan Marušič on the occasion of his 60th birthdayMarston 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: 30.12.2021; Views: 1073; Downloads: 17 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 birthdayPrimož 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: 30.12.2021; Views: 1187; Downloads: 17 Full text (370,47 KB) |
5. |
6. |
7. |
8. |
9. |
10. |