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41.
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43.
Distance-regular Cayley graphs on dihedral groups
Štefko Miklavič, Primož Potočnik, 2005, original scientific article

Abstract: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices.
Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Published in RUP: 10.07.2015; Views: 2454; Downloads: 89
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44.
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Arc-transitive cycle decompositions of tetravalent graphs
Štefko Miklavič, Primož Potočnik, Steve Wilson, 2008, original scientific article

Abstract: A cycle decomposition of a graph ▫$\Gamma$▫ is a set ▫$\mathcal{C}$▫ of cycles of ▫$\Gamma$▫ such that every edge of ▫$\Gamma$▫ belongs to exactly one cycle in ▫$\mathcal{C}$▫. Such a decomposition is called arc-transitive if the group of automorphisms of ▫$\Gamma$▫ that preserve setwise acts transitively on the arcs of ▫$\Gamma$▫. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure.
Keywords: mathematics, graph theory, cycle decomposition, automorphism group, consistent cycle, medial maps
Published in RUP: 15.10.2013; Views: 3532; Downloads: 85
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46.
Leonard triples and hypercubes
Štefko Miklavič, 2007, invited lecture at foreign university

Published in RUP: 15.10.2013; Views: 1608; Downloads: 109
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47.
On vertex-stabilizers of bipartite dual polar graphs
Štefko Miklavič, 2010, original scientific article

Abstract: Let ▫$X,Y$▫ denote vertices of a bipartite dual polar graph, and let ▫$G_X$▫ and ▫$G_Y$▫ denote the stabilizers of ▫$X$▫ and ▫$Y$▫ in the full automorphism group of this graph. In this paper, a description of the orbits of ▫$G_X \cap G_Y$▫ in the cases when the distance between ▫$X$▫ and ▫$Y$▫ is 1 or 2, is given.
Keywords: dual polar graphs, automorphism group, quadratic form, isotropic subspace
Published in RUP: 15.10.2013; Views: 2802; Downloads: 123
.pdf Full text (187,79 KB)

48.
Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups
Štefko Miklavič, Primož Šparl, 2012, original scientific article

Abstract: In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph ▫$\Gamma$▫ is ▫$n$▫-HC-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton cycle of ▫$\Gamma$▫. Similarly, ▫$\Gamma$▫ is weakly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton path of ▫$\Gamma$▫. Moreover, ▫$\Gamma$▫ is strongly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if for every such path $P$ there is a Hamilton path of ▫$\Gamma$▫ starting with ▫$P$▫. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2-HC-extendable and a complete classification of 3-HC-extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4-HP-extendable.
Keywords: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian group
Published in RUP: 15.10.2013; Views: 2726; Downloads: 143
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49.
Rotary polygons in configurations
Marko Boben, Štefko Miklavič, Primož Potočnik, 2011, original scientific article

Keywords: konfiguracija, poligon, avtomorfizem, antiavtomorfizem
Published in RUP: 15.10.2013; Views: 3214; Downloads: 142
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50.
On bipartite Q-polynominal distance-regular graphs
Štefko Miklavič, 2007, original scientific article

Abstract: Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with vertex set ▫$X$▫, diameter ▫$d \ge 3$▫ and valency ▫$k \ge 3$▫. Let ▫${\mathbb{R}}^X$▫ denote the vector space over ▫$\mathbb{R}$▫ consisting of column vectors with entries in ▫$\mathbb{r}$▫ and rows indexed by ▫$X$▫. For ▫$z \in X$▫, let ▫$\hat{z}$▫ denote the vector in ▫${\mathbb{R}}^X$▫ with a 1 in the ▫$z$▫-coordinate, and 0 in all other coordinates. Fix ▫$x,y \in X$▫ such that ▫$\partial(x,y)=2▫, where ▫$\partial$▫ denotes the path-length distance. For ▫$0 \le i,j \le d$▫ define ▫$w_{ij} = \sum\hat{z}$▫, where the sum is over all ▫$z \in X$▫ such that ▫$\partial(x,z) = i$▫ and ▫$\partial(y,z) = j▫$. We define ▫$W = \textrm{span} \{w_{ij}|0 \le i,j \le d\}$▫. In this paper we consider the space ▫$MW = \textrm{span} \{mw |m \in M, w \in W \l\}$▫, where ▫$M$▫ is the Bose-Mesner algebra of ▫$\Gamma$▫. We observe that ▫$MW$▫ is the minimal ▫$A$▫-invariant subspace of ▫${\mathbb{R}}^X$▫ which contains ▫$W$▫, where ▫$A$▫ is the adjacency matrix of ▫$\Gamma$▫. We display a basis for ▫$MW$▫ that is orthogonal with respect to the dot product. We give the action of ▫$A$▫ on this basis. We show that the dimension of ▫$MW$▫ is ▫$3d-3$▫ if ▫$\Gamma$▫ is 2-homogeneous, ▫$3d-1$▫ if ▫$\Gamma$▫ is the antipodal quotient of the ▫$2d$▫-cube, and ▫$4d-4$▫ otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the ▫$Q$▫-polynomial property.
Keywords: mathematics, graph theory, distance-regular graphs, ▫$Q$▫-polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Published in RUP: 15.10.2013; Views: 3544; Downloads: 28
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