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1.
On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1
Štefko Miklavič, 2007, original scientific article

Abstract: 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 ▫$d=4$▫.
Found in: ključnih besedah
Summary of found: ...mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions...
Keywords: mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions
Published: 15.10.2013; Views: 1687; Downloads: 14
URL Full text (0,00 KB)

2.
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.
Found in: ključnih besedah
Summary of found: ...Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with vertex set ▫$X$▫, diameter...
Keywords: mathematics, graph theory, distance-regular graphs, ▫$Q$▫-polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Published: 15.10.2013; Views: 1547; Downloads: 11
URL Full text (0,00 KB)

3.
On bipartite Q-polynomial distance-regular graphs with c[sub]2 le 2
Štefko Miklavič, Safet Penjić, 2014, original scientific article

Found in: ključnih besedah
Summary of found: ...bipartite distance-regular graph, Q-polynomial property, equitable partition, ...
Keywords: bipartite distance-regular graph, Q-polynomial property, equitable partition
Published: 15.10.2015; Views: 1053; Downloads: 28
URL Full text (0,00 KB)

4.
Ownership restructuring in transition societies from historic perspective
Andrej Bertoncelj, Matjaž Nahtigal, 2013, original scientific article

Found in: ključnih besedah
Summary of found: ...private property, mass privatization, historical background, regulatory legal framework,...
Keywords: private property, mass privatization, historical background, regulatory legal framework, transition societies, Arab spring, Mediterranean countries
Published: 21.12.2015; Views: 1209; Downloads: 10
URL Full text (0,00 KB)

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Reachability relations in digraphs
Norbert Seifter, Boris Zgrablić, Aleksander Malnič, Primož Šparl, Dragan Marušič, 2008, original scientific article

Abstract: In this paper we study reachability relations on vertices of digraphs, informally defined as follows. First, the weight of a walk is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed in the direction opposite to their orientation. Then, a vertex ▫$u$▫ is ▫$R_k^+$▫-related to a vertex ▫$v$▫ if there exists a 0-weighted walk from ▫$u$▫ to ▫$v$▫ whose every subwalk starting at u has weight in the interval ▫$[0,k]$▫. Similarly, a vertex ▫$u$▫ is ▫$R_k^-$▫-related to a vertex ▫$v$▫ if there exists a 0-weighted walk from ▫$u$▫ to ▫$v$▫ whose every subwalk starting at ▫$u$▫ has weight in the interval ▫$[-k,0]$▫. For all positive integers ▫$k$▫, the relations ▫$R_k^+$▫ and ▫$R_k^-$▫ are equivalence relations on the vertex set of a given digraph. We prove that, for transitive digraphs, properties of these relations are closely related to other properties such as having property ▫$\mathbb{Z}$▫, the number of ends, growth conditions, and vertex degree.
Found in: ključnih besedah
Summary of found: ...related to other properties such as having property ▫$\mathbb{Z}$▫, the number of ends, growth conditions,...
Keywords: graph theory, digraph, reachability relations, end of a graph, property ▫$\mathbb{Z}$▫, growth
Published: 03.04.2017; Views: 834; Downloads: 76
URL Full text (0,00 KB)

7.
On bipartite Q-polynomial distance-regular graphs with diameter 9, 10, or 11
Štefko Miklavič, 2018, original scientific article

Found in: ključnih besedah
Summary of found: ...bipartite distance-regular graph, Q-polynomial property, isomorphism...
Keywords: bipartite distance-regular graph, Q-polynomial property, isomorphism
Published: 12.03.2018; Views: 1079; Downloads: 18
URL Full text (0,00 KB)

8.
Cyclic groups are CI-groups for balanced configurations
Mikhail Muzychuk, Dragan Marušič, István Kovács, Sergio Hiroki Koike Quintanar, 2018, original scientific article

Found in: ključnih besedah
Summary of found: ...Cayley object, CI- property, cyclic group, balanced configuration...
Keywords: Cayley object, CI-property, cyclic group, balanced configuration
Published: 20.07.2018; Views: 890; Downloads: 151
URL Full text (0,00 KB)
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9.
O Terwilligerjevi algebri dvodelnih razdaljno-regularnih grafov
Safet Penjić, 2019, doctoral dissertation

Found in: ključnih besedah
Summary of found: ...stance-regular graph, Terwilliger algebra, subconstituent algebra, Q-polynomial property, equitable partition...
Keywords: bipartite distance-regular graph, Terwilliger algebra, subconstituent algebra, Q-polynomial property, equitable partition
Published: 04.06.2019; Views: 328; Downloads: 7
URL Full text (0,00 KB)

10.
On the König deficiency of zero-reducible graphs
Miklós Bartha, Miklós Ferenz Krész, 2019, original scientific article

Found in: ključnih besedah
Summary of found: ...graph matching, independent set, König property, graph reduction, graph algorithm...
Keywords: graph matching, independent set, König property, graph reduction, graph algorithm
Published: 14.11.2019; Views: 66; Downloads: 6
URL Full text (0,00 KB)

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