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1.
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: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: distance-regular graph, negative type, Terwilliger algebra
Objavljeno: 15.10.2013; Ogledov: 1655; Prenosov: 75
URL Polno besedilo (0,00 KB)

2.
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: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions
Objavljeno: 15.10.2013; Ogledov: 1772; Prenosov: 15
URL Polno besedilo (0,00 KB)

3.
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: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graph
Objavljeno: 15.10.2013; Ogledov: 1898; Prenosov: 53
URL Polno besedilo (0,00 KB)

4.
The A-like matrices for a Hypercubes
Štefko Miklavič, Paul Terwilliger, 2011, izvirni znanstveni članek

Najdeno v: ključnih besedah
Povzetek najdenega: ...A-like matrices, distance-regular graphs, Hypercubes, ...
Ključne besede: A-like matrices, distance-regular graphs, Hypercubes
Objavljeno: 15.10.2013; Ogledov: 1656; Prenosov: 27
URL Polno besedilo (0,00 KB)
Gradivo ima več datotek! Več...

5.
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: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: mathematics, graph theory, adjacency matrix, distance-regular graph, Terwilliger algebra
Objavljeno: 15.10.2013; Ogledov: 1509; Prenosov: 9
URL Polno besedilo (0,00 KB)

6.
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: ključnih besedah
Povzetek najdenega: ...mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra...
Ključne besede: mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra
Objavljeno: 15.10.2013; Ogledov: 1561; Prenosov: 61
URL Polno besedilo (0,00 KB)

7.
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: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Objavljeno: 15.10.2013; Ogledov: 1346; Prenosov: 50
URL Polno besedilo (0,00 KB)

8.
Large sets of long distance equienergetic graphs
Dragan Stevanović, 2009, izvirni znanstveni članek

Opis: Distance energy of a graph is a recent energy-type invariant, defined as the absolute deviation of the eigenvalues of the distance matrix of the graph. Two graphs of the same order are said to be distance equienergetic if they have equal distance energy, while they have distinct spectra of their distance matrices. Examples of pairs of distance equienergetic graphs appear in the literature already, but most of them have diameter two only. We describe here the distance spectrum of a special composition of regular graphs, and, as an application, we show that for any ▫$n \ge 3$▫, there exists a set of ▫$n + 1$▫ distance equienergetic graphs which have order ▫$6n$▫ and diameter ▫$n - 1$▫ each.
Najdeno v: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: graph theory, distance spectrum, distance energy, join, regular graphs
Objavljeno: 15.10.2013; Ogledov: 1483; Prenosov: 63
URL Polno besedilo (0,00 KB)

9.
The status of a rare phylogeographic lineage of the vulnerable European souslik Spermophilus citellus, endemic to central Macedonia
Boris Kryštufek, Peter Glasnović, Svetozar Petkovski, 2012, kratki znanstveni prispevek

Opis: The conversion of grasslands for agriculture has triggered a serious decline of the European ground squirrel or souslik Spermophilus citellus, categorized as Vulnerable on the IUCN Red List since 1996. The Jakupica phylogeographic lineage of central Macedonia is the smallest of the three major evolutionary lines of the European souslik. This lineage is an important reservoir of within-species diversity and should be regarded as an independent unit for conservation management purposes. It is endemic to Mount Jakupica, where it lives in mountain pastures at 1,500-2,250m altitude. The total area occupied by sousliks (884 ha) is fragmented and 94% of individuals occur in four colonies. Densities (0.8-5.5 adults ha-1) are lower than those reported elsewhere for the species, with the total population probably ,2,000 adults. One large colony, reportedly of c. 1,000 sousliks, was decimated in 2007 by a catastrophic fire and had still not recovered bz 2010. A steady decline in livestock grazing, together with the predicted advance of the tree line as a consequence of climate change, will probably reduce the optimal habitat for the souslik and negatively affect population fitness. Monitoring needs to be implemented, at least for the largest colonies, to provide early warning of any declines
Najdeno v: ključnih besedah
Povzetek najdenega: ...Spermophilus citellus, souslik, European souslik, density, endemic, distance sampling, fragmentation, Macedonia, ...
Ključne besede: Spermophilus citellus, souslik, European souslik, density, endemic, distance sampling, fragmentation, Macedonia
Objavljeno: 15.10.2013; Ogledov: 2080; Prenosov: 43
URL Polno besedilo (0,00 KB)

10.
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: ključnih besedah
Povzetek najdenega: Zadetek v naslovu
Ključne besede: graph, strongy distance-balanced, generalized Petersen graph
Objavljeno: 15.10.2013; Ogledov: 1491; Prenosov: 65
URL Polno besedilo (0,00 KB)

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