1. On bipartite Qpolynomial distanceregular graphs with c [sub] 2 [equal] 1Štefko Miklavič, 2007, original scientific article Abstract: Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫polynomial distanceregular 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 ▫$4d4$▫ cells. We use this partition to show that the intersection numbers of ▫$\Gamma$▫ satisfy the following divisibility conditions: (I) ▫$c_{i+1}1$▫ divides ▫$c_i(c_i1)$▫ for ▫$2 \le i \le d1$▫, and (II) ▫$b_{i1}1$▫ divides ▫$b_i(b_i1)$▫ for ▫$1 \le i \le d1$▫. Using these divisibility conditions we show that ▫$\Gamma$▫ does not exist if ▫$d=4$▫. Found in: ključnih besedah Keywords: mathematics, grah theory, distanceregular graphs, ▫$Q$▫polynomial property, equitable partitions Published: 15.10.2013; Views: 1694; Downloads: 14 Full text (0,00 KB) 
2. Distancebalanced graphs: Symmetry conditionsKlavdija Kutnar, Aleksander Malnič, 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 Keywords: graph theory, graph, distancebalanced graphs, vertextransitive, semysimmetric, generalized Petersen graph Published: 15.10.2013; Views: 1842; Downloads: 46 Full text (0,00 KB) 
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4. 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 Keywords: graph theory, connected graphs, connectivity, distancebalanced graphs, bipartite graphs Published: 15.10.2013; Views: 1308; Downloads: 47 Full text (0,00 KB) 
5. 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 Keywords: graph theory, distance spectrum, distance energy, join, regular graphs Published: 15.10.2013; Views: 1442; Downloads: 59 Full text (0,00 KB) 
6. On bipartite Qpolynominal distanceregular graphsŠtefko Miklavič, 2007, original scientific article Abstract: Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫polynomial distanceregular 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 pathlength 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 BoseMesner 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 ▫$3d3$▫ if ▫$\Gamma$▫ is 2homogeneous, ▫$3d1$▫ if ▫$\Gamma$▫ is the antipodal quotient of the ▫$2d$▫cube, and ▫$4d4$▫ otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the ▫$Q$▫polynomial property. Found in: ključnih besedah Keywords: mathematics, graph theory, distanceregular graphs, ▫$Q$▫polynominal property, BoseMesner algebra, balanced set characterization of the Qpolynominal property Published: 15.10.2013; Views: 1551; Downloads: 11 Full text (0,00 KB) 
7. On maximal distances in a commuting graphBojan Kuzma, Polona Oblak, Gregor Dolinar, 2012, original scientific article Abstract: It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be nonderogatory. Rankone matrices and diagonalizable matrices are also characterized in terms of the commuting graph. Found in: ključnih besedah Summary of found: ...graph, matrix algebra, algebraically closed field, centralizer, distance in graphs, ... Keywords: matematika, linearna algebra, teorija grafov, komutirajoči grafi, matrična algebra, algebraično zaprt obseg, centralizator, razdalja v grafih, mathematics, linear algebra, graph theory, commuting graph, matrix algebra, algebraically closed field, centralizer, distance in graphs Published: 03.04.2017; Views: 706; Downloads: 95 Full text (0,00 KB) This document has more files! More...

8. On the Terwilliger algebra of bipartite distanceregular graphs with [Delta][sub]2 = 0 and c[sub]2=1Mark MacLean, Štefko Miklavič, Safet Penjić, 2016, original scientific article Abstract: Let ▫$\Gamma$▫ denote a bipartite distanceregular graph with diameter ▫$D \geq 4$▫ and valency ▫$k \geq 3$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫, and let ▫$A$▫ denote the adjacency matrix of ▫$\Gamma$▫. For ▫$x \in X$▫ and for ▫$0 \leq i \leq D$▫, let ▫$\operatorname{\Gamma}_i(x)$▫ denote the set of vertices in ▫$X$▫ that are distance ▫$i$▫ from vertex ▫$x$▫. Define a parameter ▫$\operatorname{\Delta}_2$▫ in terms of the intersection numbers by ▫$\operatorname{\Delta}_2 = (k  2)(c_3  1) (c_2  1) p_{22}^2$▫. We first show that ▫$\operatorname{\Delta}_2 = 0$▫ implies that ▫$D \leq 5$▫ or ▫$c_2 \in \{1, 2 \}$▫. For ▫$x \in X$▫ let ▫$T = T(x)$▫ denote the subalgebra of ▫$\text{Mat}_X(\mathbb{C})$▫ generated by ▫$A, E_0^\ast, E_1^\ast, \ldots, E_D^\ast$▫, where for ▫$0 \leq i \leq D$, $E_i^\ast$▫ represents the projection onto the▫ $i$▫th subconstituent of ▫$\Gamma$▫ with respect to ▫$x$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. By the endpoint of an irreducible ▫$T$▫module ▫$W$▫ we mean ▫$\min \{i  E_i^\ast W \ne 0 \}$▫. In this paper we assume ▫$\Gamma$▫ has the property that for ▫$2 \leq i \leq D  1$▫, there exist complex scalars ▫$\alpha_i$▫, ▫$\beta_i$▫ such that for all ▫$x, y, z \in X$▫ with ▫$\partial(x, y) = 2$▫, ▫$\partial(x, z) = i$▫, ▫$\partial(y, z) = i$▫, we have ▫$\alpha_i + \beta_i  \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i  1}(z)  =  \operatorname{\Gamma}_{i  1}(x) \cap \operatorname{\Gamma}_{i  1}(y) \cap \operatorname{\Gamma}_1(z) $▫. We additionally assume that▫ $\operatorname{\Delta}_2 = 0$▫ with ▫$c_2 = 1$▫. Under the above assumptions we study the algebra ▫$T$▫. We show that if ▫$\Gamma$▫ is not almost 2homogeneous, then up to isomorphism there exists exactly one irreducible ▫$T$▫module with endpoint 2. We give an orthogonal basis for this ▫$T$▫module, and we give the action of ▫$A$▫ on this basis. Found in: ključnih besedah Keywords: distanceregular graphs, terwilliger algebra, subconstituent algebra Published: 14.11.2017; Views: 526; Downloads: 40 Full text (0,00 KB) 
9. On some properties of quasidistancebalanced graphsAdemir Hujdurović, 2018, original scientific article Found in: ključnih besedah Keywords: distancebalanced graphs, quasidistancebalanced graphs, bipartite graphs, bridge, razdaljnouravnoteženi grafi, kvazirazdaljnouravnoteženi grafi, dvodelni grafi, most Published: 07.02.2018; Views: 1346; Downloads: 70 Full text (0,00 KB) 
10. On the Terwilliger algebra of bipartite distanceregular graphs with $G_{i1,i1,1}(x, y, z) = alpha_i + beta_i G_{1,1,i%1}(x, y, z)$Safet Penjić, 2018, unpublished conference contribution Found in: ključnih besedah Keywords: distanceregular graphs, Terwilliger algebra, irreducible modules Published: 17.09.2018; Views: 284; Downloads: 3 Full text (0,00 KB) 