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11.
Leonard triples and hypercubes
Štefko Miklavič, 2007, izvirni znanstveni članek

Opis: Let ▫$V$▫ denote a vector space over ▫$\mathbb{C}$▫ with finite positive dimension. By a Leonard triple on ▫$V$▫ we mean an ordered triple of linear operators on ▫$V$▫ such that for each of these operators there exists a basis of ▫$V$▫ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let ▫$D$▫ denote a positive integer and let ▫${\mathcal{Q}}_D$▫ denote the graph of the ▫$D$▫-dimensional hypercube. Let ▫$X$ denote the vertex set of ▫${\mathcal{Q}}_D$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫${\mathcal{Q}}_D$▫. Fix ▫$x \in X$▫ and let ▫$A^\ast \in {\mathrm{Mat}}_X({\mathbb{C}})$▫ denote the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫${\mathrm{Mat}}_X({\mathbb{C}})$ generated by ▫$A,A^\ast$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫${\mathcal{Q}}_D$▫ with respect to ▫$x$▫. The matrices ▫$A$▫ and ▫$A^\ast$▫ are related by the fact that ▫$2iA = A^\ast A^\varepsilon - A^\varepsilon A^\ast$▫ and ▫$2iA^\ast = A^\varepsilon A - AA^\varepsilon$▫, where ▫$2iA^\varepsilon = AA^\ast - A^\ast A$▫ and ▫$i^2 = -1$▫. We show that the triple ▫$A$▫, ▫$A^\ast$▫, ▫$A^\varepsilon$▫ acts on each irreducible ▫$T$▫-module as a Leonard triple. We give a detailed description of these Leonard triples.
Najdeno v: osebi
Ključne besede: mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra
Objavljeno: 15.10.2013; Ogledov: 1625; Prenosov: 67
URL Polno besedilo (0,00 KB)

12.
On the connectivity of bipartite distance-balanced graphs
Štefko Miklavič, Primož Šparl, 2012, izvirni znanstveni članek

Opis: A connected graph ▫$\varGamma$▫ is said to be distance-balanced whenever for any pair of adjacent vertices ▫$u,v$▫ of ▫$\varGamma$▫ the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. In [K. Handa, Bipartite graphs with balanced ▫$(a,b)$▫-partitions, Ars Combin. 51 (1999), 113-119] Handa asked whether every bipartite distance-balanced graph, that is not a cycle, is 3-connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distance-balanced graph, that is not a cycle and is not 3-connected, has 18 vertices and is unique. In addition, we give a complete classification of non-3-connected bipartite distance-balanced graphs for which the minimal distance between two vertices in a 2-cut is three. All such graphs are regular and for each ▫$k \geq 3$▫ there exists an infinite family of such graphs which are ▫$k$▫-regular.Furthermore, we determine a number of structural properties that a bipartite distance-balanced graph, which is not 3-connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distance-balanced graphs.
Najdeno v: osebi
Ključne besede: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Objavljeno: 15.10.2013; Ogledov: 1447; Prenosov: 55
URL Polno besedilo (0,00 KB)

13.
Equistable graphs: conjectures, results, and connections with Boolean functions
Martin Milanič, Vadim E. Levit, Štefko Miklavič, James B. Orlin, Gábor Rudolf, D. Tankus, 2013, objavljeni povzetek znanstvenega prispevka na konferenci

Najdeno v: osebi
Ključne besede: Boolean functions
Objavljeno: 15.10.2013; Ogledov: 1844; Prenosov: 14
URL Polno besedilo (0,00 KB)

14.
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: 1524; Prenosov: 74
URL Polno besedilo (0,00 KB)

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Strongly regular tri-Cayley graphs
Klavdija Kutnar, Dragan Marušič, Štefko Miklavič, Primož Šparl, 2009, izvirni znanstveni članek

Opis: A graph is called tri-Cayley if it admits a semiregular subgroup of automorphisms having three orbits of equal length. In this paper, the structure of strongly regular tri-Cayley graphs is investigated. A structural description of strongly regular tri-Cayley graphs of cyclic groups is given.
Najdeno v: osebi
Ključne besede: strongly regular graph, tri-Cayley graph
Objavljeno: 15.10.2013; Ogledov: 1365; Prenosov: 57
URL Polno besedilo (0,00 KB)

18.
Distance-regular Cayley graphs on dihedral groups
Štefko Miklavič, Primož Potočnik, 2007, izvirni znanstveni članek

Opis: 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.
Najdeno v: osebi
Ključne besede: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Objavljeno: 15.10.2013; Ogledov: 1327; Prenosov: 63
URL Polno besedilo (0,00 KB)

19.
On Hamiltonicity of circulant digraphs of outdegree three
Štefko Miklavič, Primož Šparl, 2009, izvirni znanstveni članek

Opis: This paper deals with Hamiltonicity of connected loopless circulant digraphs of outdegree three with connection set of the form ▫$\{a,ka,c\}$▫, where ▫$k$▫ is an integer. In particular, we prove that if ▫$k=-1$▫ or ▫$k=2$▫ such a circulant digraph is Hamiltonian if and only if it is not isomorphic to the circulant digraph on 12 vertices with connection set ▫$\{3,6,4\}$▫.
Najdeno v: osebi
Ključne besede: graph theory, circulant digraph, Hamilton cycle
Objavljeno: 15.10.2013; Ogledov: 1366; Prenosov: 57
URL Polno besedilo (0,00 KB)

20.
On bipartite Q-polynominal distance-regular graphs
Štefko Miklavič, 2007, izvirni znanstveni članek

Opis: 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.
Najdeno v: osebi
Ključne besede: mathematics, graph theory, distance-regular graphs, ▫$Q$▫-polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Objavljeno: 15.10.2013; Ogledov: 1659; Prenosov: 12
URL Polno besedilo (0,00 KB)

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