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1.
Distance-balanced graphs: Symmetry conditions
Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2006, izvirni znanstveni članek

Opis: A graph ▫$X$▫ is said to be distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫, the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. A graph ▫$X$▫ is said to be strongly distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫ and any integer ▫$k$▫, the number of vertices at distance ▫$k$▫ from ▫$u$▫ and at distance ▫$k+1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$k+1$▫ from ▫$u$▫ and at distance ▫$k$▫ from ▫$v$▫. Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: GP▫$(n,2)$▫, GP▫$(5k+1,k)$▫, GP▫$(3k 3,k)$▫, and GP▫$(2k+2,k)$▫.
Najdeno v: osebi
Ključne besede: graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graph
Objavljeno: 15.10.2013; Ogledov: 1779; Prenosov: 44
URL Polno besedilo (0,00 KB)

2.
On non-normal arc-transitive 4-valent dihedrants
István Kovács, Boštjan Kuzman, Aleksander Malnič, 2010, izvirni znanstveni članek

Opis: Let ▫$X$▫ be a connected non-normal 4-valent arc-transitive Cayley graph on a dihedral group ▫$D_n$▫ such that ▫$X$▫ is bipartite, with the two bipartition sets being the two orbits of the cyclic subgroup within ▫$D_n$▫. It is shown that ▫$X$▫ is isomorphic either to the lexicographic product ▫$C_n[2K_1]$▫ with ▫$n \geq 4$▫ even, or to one of the five sporadic graphs on 10, 14, 26, 28 and 30 vertices, respectively.
Najdeno v: osebi
Ključne besede: Cayley graph, arc transitivity, dihedral group
Objavljeno: 15.10.2013; Ogledov: 1708; Prenosov: 36
URL Polno besedilo (0,00 KB)

3.
Classification of 2-arc-transitive dihedrants
Shao Fei Du, Aleksander Malnič, Dragan Marušič, 2008, izvirni znanstveni članek

Opis: A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs: (i) cycles ▫$C_{2n},\; n \ge 3$▫; (ii) complete graphs ▫$K_{2n}, \; n \ge 3$▫; (iii) complete bipartite graphs ▫$K_{n,n}, \; n \ge 3$▫; (iv) complete bipartite graphs minus a matching ▫$K_{n,n} - nK_2, \; n \ge 3$▫; (v) incidence and nonincidence graphs ▫$B(H_{11})$▫ and ▫$B'(H_{11})$▫ of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs ▫$B(PG(d,q))$▫ and ▫$B'(PG(d,q))$▫, with ▫$d \ge 2$▫ and ▫$q$▫ a prime power, of projective spaces; (vii) and an infinite family of regular ▫${\mathbb{Z}}_d$▫-covers ▫$K_{q+1}^{2d}$▫ of ▫$K_{q+1, q+1} - (q+1)K_2$▫, where ▫$q \ge 3$▫ is an odd prime power and ▫$d$▫ is a divisor of ▫$\frac{q-1}{2}$▫ and ▫$q-1$▫, respectively, depending on whether ▫$q \equiv 1 \pmod{4}$▫ or ▫$q \equiv 3 \pmod{4}$▫ obtained by identifying the vertex set of the base graph with two copies of the projective line ▫$PG(1,q)$▫, where the missing matching consists of all pairs of the form ▫$[i,i']$▫, ▫$i \in PG(1,q)$▫, and the edge ▫$[i,j']$▫ carries trivial voltage if ▫$i=\infty$▫ or ▫$j=\infty$▫, and carries voltage ▫$\bar{h} \in {\mathbb{Z}}_d$▫, the residue class of ▫$h \in {\mathbb{Z}}_d$▫, if and only if ▫$i-j = \theta^h$▫, where ▫$\theta$▫ generates the multiplicative group ▫${\mathbb{F}}_q^\ast$▫ of the Galois field ▫${\mathbb{F}}_q$▫.
Najdeno v: osebi
Ključne besede: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph
Objavljeno: 15.10.2013; Ogledov: 1520; Prenosov: 44
URL Polno besedilo (0,00 KB)

4.
Crosscovers
Aleksander Malnič, Stephen Wilson, 2010, objavljeni povzetek znanstvenega prispevka na konferenci

Najdeno v: osebi
Ključne besede: crosscovers
Objavljeno: 15.10.2013; Ogledov: 1337; Prenosov: 18
URL Polno besedilo (0,00 KB)

5.
Characterization of edge-transitive 4-valent bicirculants
István Kovács, Boštjan Kuzman, Aleksander Malnič, Stephen Wilson, 2012, izvirni znanstveni članek

Opis: Bicirkulant je graf, ki dopušča avtomorfizem z natanko dvema orbitama vozlišč enake velikosti. V članku so karakterizirani vsi neizomorfni 4-valentni povezavno tranzitivni bicirkulanti. Posledično je izpeljana karakterizacija 4-valentnih ločno tranzitivnih dihedrantov.
Najdeno v: osebi
Ključne besede: matematika, teorija grafov, štirivalenten graf, bicirkulantni graf, Cayleyev graf, povezavno tranzitiven graf, ločno tranzitiven graf, dihedrant, rose window graf, grupa avtomorfizmov
Objavljeno: 15.10.2013; Ogledov: 1939; Prenosov: 61
URL Polno besedilo (0,00 KB)

6.
Semiovals contained in the union of three concurrent lines
Aart Blokhuis, György Kiss, István Kovács, Aleksander Malnič, Dragan Marušič, János Ruff, 2007, izvirni znanstveni članek

Opis: Semiovals which are contained in the union of three concurrent lines are studied. The notion of a strong semioval is introduced, and a complete classification of these objects in PG▫$(2,p)$▫ and PG▫$(2,p^2)$▫, ▫$p$▫ an odd prime, is given.
Najdeno v: osebi
Ključne besede: mathematics, semioval, group factorization
Objavljeno: 15.10.2013; Ogledov: 1223; Prenosov: 54
URL Polno besedilo (0,00 KB)

7.
The strongly distance-balanced property of the generalized Petersen graphs
Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2009, izvirni znanstveni članek

Opis: A graph ▫$X$▫ is said to be strongly distance-balanced whenever for any edge ▫$uv$▫ of ▫$X$▫ and any positive integer ▫$i$▫, the number of vertices at distance ▫$i$▫ from ▫$u$▫ and at distance ▫$i + 1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$i + 1$▫ from ▫$u$▫ and at distance ▫$i$▫ from ▫$v$▫. It is proven that for any integers ▫$k \ge 2$▫ and ▫$n \ge k^2 + 4k + 1$▫, the generalized Petersen graph GP▫$(n, k)$▫ is not strongly distance-balanced.
Najdeno v: osebi
Ključne besede: graph, strongy distance-balanced, generalized Petersen graph
Objavljeno: 15.10.2013; Ogledov: 1434; Prenosov: 58
URL Polno besedilo (0,00 KB)

8.
Covering space techniques in graph theory
Aleksander Malnič, 2010, objavljeni povzetek znanstvenega prispevka na konferenci

Najdeno v: osebi
Ključne besede: graph theory, graph covers, lifting automorphisms
Objavljeno: 15.10.2013; Ogledov: 1836; Prenosov: 27
URL Polno besedilo (0,00 KB)
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9.
Hamiltonicity of cubic Cayley graphs
Dragan Marušič, Henry Glover, Klavdija Kutnar, Aleksander Malnič, 2012, objavljeni povzetek znanstvenega prispevka na konferenci (vabljeno predavanje)

Najdeno v: osebi
Ključne besede: Cayley graph, Hamilton path, Hamilton cycle, arc-transitive graph, Cayley map
Objavljeno: 15.10.2013; Ogledov: 1329; Prenosov: 23
URL Polno besedilo (0,00 KB)

10.
On 2-fold covers of graphs
Yan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, izvirni znanstveni članek

Opis: A regular covering projection ▫$\wp : \widetilde{X} \to X$▫ of connected graphs is ▫$G$▫-admissible if ▫$G$▫ lifts along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of covering transformations. The projection is called ▫$G$▫-split whenever the extension ▫{$\mathrm{CT}}(\wp) \to \tilde{G} \to G$▫ splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that ▫$G$▫ is transitive on ▫$X$▫, a ▫$G$▫-split cover is said to be ▫$G$▫-split-transitive if all complements ▫$\tilde{G} \cong G$▫ of CT▫$(\wp)$▫ within ▫$\tilde{G}$▫ are transitive on ▫$\widetilde{X}$▫; it is said to be ▫$G$▫-split-sectional whenever for each complement ▫$\tilde{G}$▫ there exists a ▫$\tilde{G}$▫-invariant section of ▫$\wp$▫; and it is called ▫$G$▫-split-mixed otherwise. It is shown, when ▫$G$▫ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. Split-transitive covers, however, are considerably more difficult to analyze. For cubic symmetric graphs split 2-cover are necessarily canonical double covers (that is, no ▫$G$▫-split-transitive 2-covers exist) when ▫$G$▫ is 1-regular or 4-regular. In all other cases, that is, if ▫$G$▫ is ▫$s$▫-regular, ▫$s=2,3$▫ or ▫$5$▫, a necessary and sufficient condition for the existence of a transitive complement ▫$\tilde{G}$▫ is given, and moreover, an infinite family of split-transitive 2-covers based on the alternating groups of the form ▫$A_{12k+10}$▫ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group ▫$G$▫ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.
Najdeno v: osebi
Ključne besede: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Objavljeno: 15.10.2013; Ogledov: 1191; Prenosov: 13
URL Polno besedilo (0,00 KB)

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