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31.
The V Latin-American Algorithms, Graphs, and Optimization Symposium
2009, proceedings of peer-reviewed scientific conference contributions (international and foreign conferences)

Keywords: algorithms, graph theory, mathematical optimization, international conferences
Published in RUP: 15.10.2013; Views: 2868; Downloads: 80
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32.
Covering space techniques in graph theory
Aleksander Malnič, 2010, published scientific conference contribution abstract

Keywords: graph theory, graph covers, lifting automorphisms
Published in RUP: 15.10.2013; Views: 3646; Downloads: 87
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33.
On cubic biabelian graphs
István Kovács, 2010, published scientific conference contribution abstract

Keywords: graph theory
Published in RUP: 15.10.2013; Views: 2900; Downloads: 28
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34.
Large sets of long distance equienergetic graphs
Dragan Stevanović, 2009, original scientific article

Abstract: 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.
Keywords: graph theory, distance spectrum, distance energy, join, regular graphs
Published in RUP: 15.10.2013; Views: 3469; Downloads: 136
.pdf Full text (144,63 KB)

35.
On the connectivity of bipartite distance-balanced graphs
Štefko Miklavič, Primož Šparl, 2012, original scientific article

Abstract: 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.
Keywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Published in RUP: 15.10.2013; Views: 3242; Downloads: 96
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36.
Leonard triples and hypercubes
Štefko Miklavič, 2007, original scientific article

Abstract: 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.
Keywords: mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra
Published in RUP: 15.10.2013; Views: 3806; Downloads: 121
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37.
Consistent Cycles in 1/2-Arc-Transitive Graphs
Marko Boben, Štefko Miklavič, Primož Potočnik, 2009, original scientific article

Keywords: mathematics, graph theory, 1/2-arc-transitivity, consistent cycle
Published in RUP: 15.10.2013; Views: 4321; Downloads: 49
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38.
A note on a conjecture on consistent cycles
Štefko Miklavič, 2013, original scientific article

Abstract: Let ▫$\Gamma$▫ denote a finite digraph and let ▫$G$▫ be a subgroup of its automorphism group. A directed cycle ▫$\vec{C}$▫ of▫ $\Gamma$▫ is called ▫$G$▫-consistent whenever there is an element of ▫$G$▫ whose restriction to▫ $\vec{C}$▫ is the 1-step rotation of ▫$\vec{C}$▫. In this short note we provea conjecture on ▫$G$▫-consistent directed cycles stated by Steve Wilson.
Keywords: graph theory, digraphs, consistent directed cycles
Published in RUP: 15.10.2013; Views: 2541; Downloads: 123
.pdf Full text (229,03 KB)

39.
On quartic half-arc-transitive metacirculants
Dragan Marušič, Primož Šparl, 2008, original scientific article

Abstract: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫-semiregular for some integers ▫$m \ge 1$▫, ▫$n \ge 2▫$, and where ▫$\sigma$▫ normalizes ▫$\rho$▫, cyclically permuting the orbits of ▫$\rho$▫ in such a way that ▫$\sigma^m$▫ has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.
Keywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group
Published in RUP: 15.10.2013; Views: 3705; Downloads: 132
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40.
Q-polynomial distance-regular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0
Štefko Miklavič, 2008, original scientific article

Abstract: 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 ▫$\Gamma$▫.
Keywords: mathematics, graph theory, adjacency matrix, distance-regular graph, Terwilliger algebra
Published in RUP: 15.10.2013; Views: 4200; Downloads: 30
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