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21.
On the recognition of k-equistable graphs
Vadim E. Levit, Martin Milanič, D. Tankus, 2012, published scientific conference contribution

Found in: ključnih besedah
Summary of found: ...maximal stable set, equistable graph, polynomial time algorithm, ...
Keywords: maximal stable set, equistable graph, polynomial time algorithm
Published: 15.10.2013; Views: 1283; Downloads: 65
URL Full text (0,00 KB)

22.
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.
Found in: ključnih besedah
Summary of found: ...positive integer and let ▫${\mathcal{Q}}_D$▫ denote the graph of the ▫$D$▫-dimensional hypercube. Let ▫$X$ denote...
Keywords: mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra
Published: 15.10.2013; Views: 1598; Downloads: 63
URL Full text (0,00 KB)

23.
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.
Found in: ključnih besedah
Summary of found: ...A connected graph ▫$\varGamma$▫ is said to be distance-balanced whenever...
Keywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Published: 15.10.2013; Views: 1400; Downloads: 52
URL Full text (0,00 KB)

24.
The full automorphism group of a Cayley graph
Gabriel Verret, 2013, published scientific conference contribution abstract (invited lecture)

Found in: ključnih besedah
Keywords: automorphism group, Cayley graph
Published: 15.10.2013; Views: 2334; Downloads: 21
URL Full text (0,00 KB)
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25.
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.
Found in: ključnih besedah
Summary of found: ...Distance energy of a graph is a recent energy-type invariant, defined as...
Keywords: graph theory, distance spectrum, distance energy, join, regular graphs
Published: 15.10.2013; Views: 1510; Downloads: 67
URL Full text (0,00 KB)

26.
On cubic biabelian graphs
István Kovács, 2010, published scientific conference contribution abstract

Found in: ključnih besedah
Summary of found: ... graph theory, ...
Keywords: graph theory
Published: 15.10.2013; Views: 1591; Downloads: 11
URL Full text (0,00 KB)

27.
The strongly distance-balanced property of the generalized Petersen graphs
Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2009, original scientific article

Abstract: 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.
Found in: ključnih besedah
Summary of found: ...A graph ▫$X$▫ is said to be strongly distance-balanced...
Keywords: graph, strongy distance-balanced, generalized Petersen graph
Published: 15.10.2013; Views: 1511; Downloads: 68
URL Full text (0,00 KB)

28.
The V Latin-American Algorithms, Graphs, and Optimization Symposium
2009, proceedings of peer-reviewed scientific conference contributions (international and foreign conferences)

Found in: ključnih besedah
Summary of found: ...algorithms, graph theory, mathematical optimization, international conferences, ...
Keywords: algorithms, graph theory, mathematical optimization, international conferences
Published: 15.10.2013; Views: 1235; Downloads: 28
URL Full text (0,00 KB)

29.
Covering space techniques in graph theory
Aleksander Malnič, 2010, published scientific conference contribution abstract

Found in: ključnih besedah
Keywords: graph theory, graph covers, lifting automorphisms
Published: 15.10.2013; Views: 1960; Downloads: 29
URL Full text (0,00 KB)
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30.
The plane-width of graphs
Marcin Kamiński, Paul Medvedev, Martin Milanič, 2011, original scientific article

Found in: ključnih besedah
Summary of found: ...plane-width, realization of a graph, packing non-overlapping unit disks, chromatic number, circular...
Keywords: plane-width, realization of a graph, packing non-overlapping unit disks, chromatic number, circular chromatic number
Published: 15.10.2013; Views: 1213; Downloads: 35
URL Full text (0,00 KB)

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