1. The Terwilliger algebra of a distanceregular graph of negative typeŠtefko Miklavič, 2009, original scientific article Abstract: Let ▫$\Gamma$▫ denote a distanceregular 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 ▫$2D2$▫. For these ▫$T$▫modules we display a basis consisting of eigenvectors for ▫$A^\ast$▫, and for each basis we give the action of ▫$A$▫. Found in: ključnih besedah Keywords: distanceregular graph, negative type, Terwilliger algebra Published: 15.10.2013; Views: 2728; Downloads: 105 Full text (0,00 KB) 
2. A complexitystudy of distance variants of covering and domination problems in Hfree graphsMirza Krbezlija, 2021, master's thesis Found in: ključnih besedah Summary of found: ...distancek vertex cover, distance kedge cover, Hfree graph, polynomialtime algorithm, NPcomplete problem, dichotomy theorem... Keywords: distancek dominating set, distancek edge dominating set, distancek vertex cover, distance kedge cover, Hfree graph, polynomialtime algorithm, NPcomplete problem, dichotomy theorem Published: 18.10.2021; Views: 451; Downloads: 15 Full text (0,00 KB) 
3. Distancebalanced graphs: Symmetry conditionsAleksander Malnič, Klavdija Kutnar, Dragan Marušič, Štefko Miklavič, 2006, original scientific article Abstract: A graph ▫$X$▫ is said to be distancebalanced 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 distancebalanced 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) distancebalanced is the main theme of this article. That a vertextransitive graph is necessarily strongly distancebalanced and thus also distancebalanced 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 edgetransitive, but not vertextransitive) which are distancebalanced, but there are also infinite families of semisymmetric graphs which are not distancebalanced. Results on the distancebalanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distancebalanced 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▫$(2k+2,k)$▫. Found in: ključnih besedah Summary of found: ...A graph ▫$X$▫ is said to be distancebalanced if... Keywords: graph theory, graph, distancebalanced graphs, vertextransitive, semysimmetric, generalized Petersen graph Published: 15.10.2013; Views: 3235; Downloads: 85 Full text (0,00 KB) 
4. Qpolynomial distanceregular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0Štefko Miklavič, 2008, original scientific article Abstract: Let ▫$\Gamma$▫ denote a ▫$Q$▫polynomial distanceregular 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 ▫$2D2$▫. 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 ▫$k1$▫, where ▫$k$▫ is the valency of ▫$\Gamma$▫. Found in: ključnih besedah Summary of found: ...Let ▫$\Gamma$▫ denote a ▫$Q$▫polynomial distanceregular graph with diameter ▫$D \ge 3$▫ and intersection... Keywords: mathematics, graph theory, adjacency matrix, distanceregular graph, Terwilliger algebra Published: 15.10.2013; Views: 3242; Downloads: 25 Full text (0,00 KB) 
5. 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: ...mathematics, graph theory, Leonard triple, distanceregular graph, hypercube, Terwilliger algebra... Keywords: mathematics, graph theory, Leonard triple, distanceregular graph, hypercube, Terwilliger algebra Published: 15.10.2013; Views: 2914; Downloads: 115 Full text (0,00 KB) 
6. On the connectivity of bipartite distancebalanced graphsŠtefko Miklavič, Primož Šparl, 2012, original scientific article Abstract: A connected graph ▫$\varGamma$▫ is said to be distancebalanced 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), 113119] Handa asked whether every bipartite distancebalanced graph, that is not a cycle, is 3connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distancebalanced graph, that is not a cycle and is not 3connected, has 18 vertices and is unique. In addition, we give a complete classification of non3connected bipartite distancebalanced graphs for which the minimal distance between two vertices in a 2cut 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 distancebalanced graph, which is not 3connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distancebalanced graphs. Found in: ključnih besedah Summary of found: ...A connected graph ▫$\varGamma$▫ is said to be distancebalanced whenever... Keywords: graph theory, connected graphs, connectivity, distancebalanced graphs, bipartite graphs Published: 15.10.2013; Views: 2560; Downloads: 92 Full text (0,00 KB) 
7. Large sets of long distance equienergetic graphsDragan Stevanović, 2009, original scientific article Abstract: Distance energy of a graph is a recent energytype 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 energytype invariant, defined as... Keywords: graph theory, distance spectrum, distance energy, join, regular graphs Published: 15.10.2013; Views: 2760; Downloads: 134 This document has more files! More...

8. The strongly distancebalanced property of the generalized Petersen graphsŠtefko Miklavič, Dragan Marušič, Aleksander Malnič, Klavdija Kutnar, 2009, original scientific article Abstract: A graph ▫$X$▫ is said to be strongly distancebalanced 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 distancebalanced. Found in: ključnih besedah Summary of found: ...A graph ▫$X$▫ is said to be strongly distancebalanced... Keywords: graph, strongy distancebalanced, generalized Petersen graph Published: 15.10.2013; Views: 2423; Downloads: 125 This document has more files! More...

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