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On the Terwilliger algebra of bipartite distance-regular graphs with [Delta][sub]2 = 0 and c[sub]2=1
Mark MacLean, Štefko Miklavič, Safet Penjić, 2016, original scientific article

Abstract: Let ▫$\Gamma$▫ denote a bipartite distance-regular 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 2-homogeneous, 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.
Keywords: distance-regular graphs, terwilliger algebra, subconstituent algebra
Published in RUP: 14.11.2017; Views: 2250; Downloads: 143
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On bipartite Q-polynominal distance-regular graphs
Štefko Miklavič, 2007, original scientific article

Abstract: Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular 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 path-length 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 Bose-Mesner 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 ▫$3d-3$▫ if ▫$\Gamma$▫ is 2-homogeneous, ▫$3d-1$▫ if ▫$\Gamma$▫ is the antipodal quotient of the ▫$2d$▫-cube, and ▫$4d-4$▫ otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the ▫$Q$▫-polynomial property.
Keywords: mathematics, graph theory, distance-regular graphs, ▫$Q$▫-polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Published in RUP: 15.10.2013; Views: 3541; Downloads: 28
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On 2-fold covers of graphs
Yan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, original scientific article

Abstract: A regular covering projection ▫$\wp : \widetilde{X} \to X$▫ of connected graphs is ▫$G$▫-admissible if ▫$G$▫ lifts along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of covering transformations. The projection is called ▫$G$▫-split whenever the extension ▫{$\mathrm{CT}}(\wp) \to \tilde{G} \to G$▫ splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that ▫$G$▫ is transitive on ▫$X$▫, a ▫$G$▫-split cover is said to be ▫$G$▫-split-transitive if all complements ▫$\tilde{G} \cong G$▫ of CT▫$(\wp)$▫ within ▫$\tilde{G}$▫ are transitive on ▫$\widetilde{X}$▫; it is said to be ▫$G$▫-split-sectional whenever for each complement ▫$\tilde{G}$▫ there exists a ▫$\tilde{G}$▫-invariant section of ▫$\wp$▫; and it is called ▫$G$▫-split-mixed otherwise. It is shown, when ▫$G$▫ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. Split-transitive covers, however, are considerably more difficult to analyze. For cubic symmetric graphs split 2-cover are necessarily canonical double covers (that is, no ▫$G$▫-split-transitive 2-covers exist) when ▫$G$▫ is 1-regular or 4-regular. In all other cases, that is, if ▫$G$▫ is ▫$s$▫-regular, ▫$s=2,3$▫ or ▫$5$▫, a necessary and sufficient condition for the existence of a transitive complement ▫$\tilde{G}$▫ is given, and moreover, an infinite family of split-transitive 2-covers based on the alternating groups of the form ▫$A_{12k+10}$▫ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group ▫$G$▫ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.
Keywords: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Published in RUP: 15.10.2013; Views: 3532; Downloads: 34
URL Link to full text

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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
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