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1.
On distance-regular Cayley graphs for abelian groups
Štefko Miklavič, 2012, predavanje na tuji univerzi

Najdeno v: osebi
Objavljeno: 15.10.2013; Ogledov: 1104; Prenosov: 45
URL Polno besedilo (0,00 KB)

2.
The Terwilliger algebra of a distance-regular graph of negative type
Štefko Miklavič, 2009, izvirni znanstveni članek

Opis: Let ▫$\Gamma$▫ denote a distance-regular graph with diameter ▫$D \ge 3$▫. Assume ▫$\Gamma$▫ has classical parameters ▫$(D,b,\alpha,\beta)▫$ with ▫$b < -1$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X(\mathbb{C})$▫ denote the adjacency matrix of ▫$\Gamma$▫. 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 call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exist exactly two irreducible ▫$T$▫-modules with endpoint 1; their dimensions are ▫$D$▫ and ▫$2D-2$▫. For these ▫$T$▫-modules we display a basis consisting of eigenvectors for ▫$A^\ast$▫, and for each basis we give the action of ▫$A$▫.
Najdeno v: osebi
Ključne besede: distance-regular graph, negative type, Terwilliger algebra
Objavljeno: 15.10.2013; Ogledov: 1622; Prenosov: 70
URL Polno besedilo (0,00 KB)

3.
On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1
Štefko Miklavič, 2007, izvirni znanstveni članek

Opis: Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with diameter ▫$d \ge 3$▫, valency ▫$k \ge 3$▫ and intersection number ▫$c_2=1$▫. We show that ▫$\Gamma$▫ has a certain equitable partition of its vertex set which involves ▫$4d-4$▫ cells. We use this partition to show that the intersection numbers of ▫$\Gamma$▫ satisfy the following divisibility conditions: (I) ▫$c_{i+1}-1$▫ divides ▫$c_i(c_i-1)$▫ for ▫$2 \le i \le d-1$▫, and (II) ▫$b_{i-1}-1$▫ divides ▫$b_i(b_i-1)$▫ for ▫$1 \le i \le d-1$▫. Using these divisibility conditions we show that ▫$\Gamma$▫ does not exist if ▫$d=4$▫.
Najdeno v: osebi
Ključne besede: mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions
Objavljeno: 15.10.2013; Ogledov: 1687; Prenosov: 14
URL Polno besedilo (0,00 KB)

4.
Consistent cycles
Štefko Miklavič, 2005, predavanje na tuji univerzi

Najdeno v: osebi
Objavljeno: 15.10.2013; Ogledov: 944; Prenosov: 14
URL Polno besedilo (0,00 KB)

5.
Leonard triples and hypercubes
Štefko Miklavič, 2006, predavanje na tuji univerzi

Najdeno v: osebi
Objavljeno: 15.10.2013; Ogledov: 1038; Prenosov: 30
URL Polno besedilo (0,00 KB)

6.
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: 1833; Prenosov: 46
URL Polno besedilo (0,00 KB)

7.
The A-like matrices for a Hypercubes
Štefko Miklavič, Paul Terwilliger, 2011, izvirni znanstveni članek

Najdeno v: osebi
Ključne besede: A-like matrices, distance-regular graphs, Hypercubes
Objavljeno: 15.10.2013; Ogledov: 1600; Prenosov: 25
URL Polno besedilo (0,00 KB)
Gradivo ima več datotek! Več...

8.
Q-polynomial distance-regular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0
Štefko Miklavič, 2008, izvirni znanstveni članek

Opis: Let ▫$\Gamma$▫ denote a ▫$Q$▫-polynomial distance-regular graph with diameter ▫$D \ge 3$▫ and intersection numbers ▫$a_1=0$▫, ▫$a_2 \ne 0$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫ and let denote $A^\ast \in {\mathrm{Mat}}_X ({\mathbb{C}})$ the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫$A{\mathrm{Mat}}_X ({\mathbb{C}})$▫ generated by ▫$A$▫, ▫$A^\ast$▫. We call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exists a unique irreducible ▫$T$▫-module ▫$W$▫ with endpoint 1. We show that ▫$W$▫ has dimension ▫$2D-2$▫. We display a basis for ▫$W$▫ which consists of eigenvectors for ▫$A^\ast$▫. We display the action of ▫$A$▫ on this basis. We show that ▫$W$▫ appears in the standard module of ▫$\Gamma$▫ with multiplicity ▫$k-1$▫, where ▫$k$▫ is the valency of ▫$\Gamma$▫.
Najdeno v: osebi
Ključne besede: mathematics, graph theory, adjacency matrix, distance-regular graph, Terwilliger algebra
Objavljeno: 15.10.2013; Ogledov: 1443; Prenosov: 9
URL Polno besedilo (0,00 KB)

9.
A note on a conjecture on consistent cycles
Štefko Miklavič, 2013, izvirni znanstveni članek

Najdeno v: osebi
Ključne besede: digraphs, consistent directed cycles
Objavljeno: 15.10.2013; Ogledov: 1028; Prenosov: 55
URL Polno besedilo (0,00 KB)

10.
Consistent Cycles in 1/2-Arc-Transitive Graphs
Marko Boben, Štefko Miklavič, Primož Potočnik, 2009, izvirni znanstveni članek

Najdeno v: osebi
Ključne besede: mathematics, graph theory, 1/2-arc-transitivity, consistent cycle
Objavljeno: 15.10.2013; Ogledov: 2136; Prenosov: 7
URL Polno besedilo (0,00 KB)
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