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1.
On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1
Štefko Miklavič, 2007, izvirni znanstveni članek

Opis: Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular 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 ▫$4d-4$▫ cells. We use this partition to show that the intersection numbers of ▫$\Gamma$▫ satisfy the following divisibility conditions: (I) ▫$c_{i+1}-1$▫ divides ▫$c_i(c_i-1)$▫ for ▫$2 \le i \le d-1$▫, and (II) ▫$b_{i-1}-1$▫ divides ▫$b_i(b_i-1)$▫ for ▫$1 \le i \le d-1$▫. Using these divisibility conditions we show that ▫$\Gamma$▫ does not exist if ▫$d=4$▫.
Najdeno v: ključnih besedah
Ključne besede: mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions
Objavljeno: 15.10.2013; Ogledov: 1635; Prenosov: 14
URL Polno besedilo (0,00 KB)

2.
The A-like matrices for a Hypercubes
Štefko Miklavič, Paul Terwilliger, 2011, izvirni znanstveni članek

Najdeno v: ključnih besedah
Povzetek najdenega: ...A-like matrices, distance- regular graphs, Hypercubes, ...
Ključne besede: A-like matrices, distance-regular graphs, Hypercubes
Objavljeno: 15.10.2013; Ogledov: 1570; Prenosov: 24
URL Polno besedilo (0,00 KB)
Gradivo ima več datotek! Več...

3.
A complete classification of cubic symmetric graphs of girth 6
Klavdija Kutnar, Dragan Marušič, 2009, izvirni znanstveni članek

Opis: A complete classification of cubic symmetric graphs of girth 6 is given. It is shown that with the exception of the Heawood graph, the Moebius-Kantor graph, the Pappus graph, and the Desargues graph, a cubic symmetric graph ▫$X$▫ of girth 6 is a normal Cayley graph of a generalized dihedral group; in particular, (i) ▫$X$▫ is 2-regular if and only if it is isomorphic to a so-called ▫$I_k^n$▫-path, a graph of order either ▫$n^2/2$▫ or ▫$n^2/6$▫, which is characterized by the fact that its quotient relative to a certain semiregular automorphism is a path. (ii) ▫$X$▫ is 1-regular if and only if there exists an integer ▫$r$▫ with prime decomposition ▫$r=3^s p_1^{e_1} \dots p_t^{e_t} > 3$▫, where ▫$s \in \{0,1\}$▫, ▫$t \ge 1$▫, and ▫$p_i \equiv 1 \pmod{3}$▫, such that ▫$X$▫ is isomorphic either to a Cayley graph of a dihedral group ▫$D_{2r}$▫ of order ▫$2r$▫ or ▫$X$▫ is isomorphic to a certain ▫$\ZZ_r$▫-cover of one of the following graphs: the cube ▫$Q_3$▫, the Pappus graph or an ▫$I_k^n(t)$▫-path of order ▫$n^2/2$▫.
Najdeno v: ključnih besedah
Povzetek najdenega: ...dihedral group; in particular, (i) ▫$X$▫ is 2- regular if and only if it is isomorphic...
Ključne besede: graph theory, cubic graphs, symmetric graphs, ▫$s$▫-regular graphs, girth, consistent cycle
Objavljeno: 15.10.2013; Ogledov: 1767; Prenosov: 43
URL Polno besedilo (0,00 KB)

4.
Large sets of long distance equienergetic graphs
Dragan Stevanović, 2009, izvirni znanstveni članek

Opis: 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.
Najdeno v: ključnih besedah
Povzetek najdenega: ...distance spectrum of a special composition of regular graphs, and, as an application, we show...
Ključne besede: graph theory, distance spectrum, distance energy, join, regular graphs
Objavljeno: 15.10.2013; Ogledov: 1410; Prenosov: 55
URL Polno besedilo (0,00 KB)

5.
On 2-fold covers of graphs
Yan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, izvirni znanstveni članek

Opis: 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.
Najdeno v: ključnih besedah
Povzetek najdenega: ...A regular covering projection ▫$\wp : \widetilde{X} \to X$▫...
Ključne besede: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Objavljeno: 15.10.2013; Ogledov: 1191; Prenosov: 13
URL Polno besedilo (0,00 KB)

6.
On bipartite Q-polynominal distance-regular graphs
Štefko Miklavič, 2007, izvirni znanstveni članek

Opis: 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.
Najdeno v: ključnih besedah
Ključne besede: mathematics, graph theory, distance-regular graphs, ▫$Q$▫-polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Objavljeno: 15.10.2013; Ogledov: 1521; Prenosov: 11
URL Polno besedilo (0,00 KB)

7.
A note on the spectral characterization of dumbbell graphs
JianFeng Wang, Qiongxiang Huang, Francesco Belardo, Enzo M. Li Marzi, 2009, izvirni znanstveni članek

Najdeno v: ključnih besedah
Povzetek najdenega: ...grafi, spektri, skoraj regularni grafi, graphs, specters, almost regular graphs, ...
Ključne besede: grafi, spektri, skoraj regularni grafi, graphs, specters, almost regular graphs
Objavljeno: 15.10.2015; Ogledov: 875; Prenosov: 58
URL Polno besedilo (0,00 KB)

8.
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, izvirni znanstveni članek

Opis: 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.
Najdeno v: ključnih besedah
Ključne besede: distance-regular graphs, terwilliger algebra, subconstituent algebra
Objavljeno: 14.11.2017; Ogledov: 494; Prenosov: 38
URL Polno besedilo (0,00 KB)

9.
Tetravalent vertex- and edge-transitive graphs over doubled cycles
Boštjan Kuzman, Aleksander Malnič, Primož Potočnik, 2018, izvirni znanstveni članek

Najdeno v: ključnih besedah
Povzetek najdenega: ...tetravalent graphs, symmetric graphs, regular covers, cyclic codes, reflexible polynomials...
Ključne besede: tetravalent graphs, symmetric graphs, regular covers, cyclic codes, reflexible polynomials
Objavljeno: 02.03.2018; Ogledov: 606; Prenosov: 69
URL Polno besedilo (0,00 KB)

10.
On the Terwilliger algebra of bipartite distance-regular graphs with $G_{i-1,i-1,1}(x, y, z) = alpha_i + beta_i G_{1,1,i%1}(x, y, z)$
Safet Penjić, 2018, prispevek na konferenci brez natisa

Najdeno v: ključnih besedah
Ključne besede: distance-regular graphs, Terwilliger algebra, irreducible modules
Objavljeno: 17.09.2018; Ogledov: 260; Prenosov: 3
URL Polno besedilo (0,00 KB)

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