21. On the recognition of k-equistable graphsVadim 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: 1404; Downloads: 73 Full text (0,00 KB) |
22. Leonard triples and hypercubesŠtefko 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: 1727; Downloads: 69 Full text (0,00 KB) |
23. On the connectivity of bipartite distance-balanced graphsŠtefko 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: 1528; Downloads: 56 Full text (0,00 KB) |
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25. Large sets of long distance equienergetic graphsDragan 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: 1630; Downloads: 76 Full text (0,00 KB) |
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27. The strongly distance-balanced property of the generalized Petersen graphsKlavdija 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: 1592; Downloads: 78 Full text (0,00 KB) |
28. The V Latin-American Algorithms, Graphs, and Optimization Symposium2009, 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: 1361; Downloads: 30 Full text (0,00 KB) |
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30. The plane-width of graphsMarcin 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: 1320; Downloads: 36 Full text (0,00 KB) |