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12. On nonnormal arctransitive 4valent dihedrantsIstván Kovács, Boštjan Kuzman, Aleksander Malnič, 2010, original scientific article Abstract: Let ▫$X$▫ be a connected nonnormal 4valent arctransitive Cayley graph on a dihedral group ▫$D_n$▫ such that ▫$X$▫ is bipartite, with the two bipartition sets being the two orbits of the cyclic subgroup within ▫$D_n$▫. It is shown that ▫$X$▫ is isomorphic either to the lexicographic product ▫$C_n[2K_1]$▫ with ▫$n \geq 4$▫ even, or to one of the five sporadic graphs on 10, 14, 26, 28 and 30 vertices, respectively. Found in: ključnih besedah Summary of found: ...be a connected nonnormal 4valent arctransitive Cayley graph on a dihedral group ▫$D_n$▫ such that... Keywords: Cayley graph, arc transitivity, dihedral group Published: 15.10.2013; Views: 1827; Downloads: 45 Full text (0,00 KB) 
13. Classification of 2arctransitive dihedrantsShao Fei Du, Aleksander Malnič, Dragan Marušič, 2008, original scientific article Abstract: A complete classification of 2arctransitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2arctransitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162196]. The list consists of the following graphs: (i) cycles ▫$C_{2n},\; n \ge 3$▫; (ii) complete graphs ▫$K_{2n}, \; n \ge 3$▫; (iii) complete bipartite graphs ▫$K_{n,n}, \; n \ge 3$▫; (iv) complete bipartite graphs minus a matching ▫$K_{n,n}  nK_2, \; n \ge 3$▫; (v) incidence and nonincidence graphs ▫$B(H_{11})$▫ and ▫$B'(H_{11})$▫ of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs ▫$B(PG(d,q))$▫ and ▫$B'(PG(d,q))$▫, with ▫$d \ge 2$▫ and ▫$q$▫ a prime power, of projective spaces; (vii) and an infinite family of regular ▫${\mathbb{Z}}_d$▫covers ▫$K_{q+1}^{2d}$▫ of ▫$K_{q+1, q+1}  (q+1)K_2$▫, where ▫$q \ge 3$▫ is an odd prime power and ▫$d$▫ is a divisor of ▫$\frac{q1}{2}$▫ and ▫$q1$▫, respectively, depending on whether ▫$q \equiv 1 \pmod{4}$▫ or ▫$q \equiv 3 \pmod{4}$▫ obtained by identifying the vertex set of the base graph with two copies of the projective line ▫$PG(1,q)$▫, where the missing matching consists of all pairs of the form ▫$[i,i']$▫, ▫$i \in PG(1,q)$▫, and the edge ▫$[i,j']$▫ carries trivial voltage if ▫$i=\infty$▫ or ▫$j=\infty$▫, and carries voltage ▫$\bar{h} \in {\mathbb{Z}}_d$▫, the residue class of ▫$h \in {\mathbb{Z}}_d$▫, if and only if ▫$ij = \theta^h$▫, where ▫$\theta$▫ generates the multiplicative group ▫${\mathbb{F}}_q^\ast$▫ of the Galois field ▫${\mathbb{F}}_q$▫. Found in: ključnih besedah Summary of found: ...classification of 2arctransitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing... Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2Arctransitive graph Published: 15.10.2013; Views: 1619; Downloads: 52 Full text (0,00 KB) 
14. A complete classification of cubic symmetric graphs of girth 6Klavdija Kutnar, Dragan Marušič, 2009, original scientific article Abstract: A complete classification of cubic symmetric graphs of girth 6 is given. It is shown that with the exception of the Heawood graph, the MoebiusKantor 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 2regular if and only if it is isomorphic to a socalled ▫$I_k^n$▫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 1regular 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 ▫$n^2/2$▫. Found in: ključnih besedah Summary of found: ...A complete classification of cubic symmetric graphs of girth 6 is given. It is... Keywords: graph theory, cubic graphs, symmetric graphs, ▫$s$▫regular graphs, girth, consistent cycle Published: 15.10.2013; Views: 1851; Downloads: 53 Full text (0,00 KB) 
15. Rose window graphs underlying rotary mapsIstván Kovács, Klavdija Kutnar, János Ruff, 2010, published scientific conference contribution Abstract: Given natural numbers ▫$n \ge 3$▫ and ▫$1 \le a$▫, ▫$r \le n1$▫, the rose window graph ▫$R_n(a,r)$▫ is a quartic graph with vertex set ▫$\{x_i \vert\; i \in {\mathbb Z}_n \} \cup \{y_i \vert\; i \in {\mathbb Z}_n \}$▫ and edge set ▫$\{\{x_i, x_{i+1}\} \vert\; i \in {\mathbb Z}_n \} \cup \{\{y_i, y_{i+1}\} \vert\; i \in {\mathbb Z}_n \} \cup \{\{x_i, y_i\} \vert\; i \in {\mathbb Z}_n\} \cup \{\{x_{i+a}, y_i\} \vert\; i \in {\mathbb Z}_n \}$▫. In this paper rotary maps on rose window graphs are considered. In particular, we answer the question posed in [S. Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 719. http://amc.imfm.si/index.php/amc/issue/view/5] concerning which of these graphs underlie a rotary map. Found in: ključnih besedah Summary of found: ...a$▫, ▫$r \le n1$▫, the rose window graph ▫$R_n(a,r)$▫ is a quartic graph with vertex... Keywords: graph theory, rotary map, edgetransitive graph, covering graph, voltage graph Published: 15.10.2013; Views: 1614; Downloads: 51 Full text (0,00 KB) 
16. Qpolynomial distanceregular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0Štefko Miklavič, 2008, original scientific article Abstract: Let ▫$\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: 1567; Downloads: 9 Full text (0,00 KB) 
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18. On quartic halfarctransitive metacirculantsDragan Marušič, Primož Šparl, 2008, original scientific article Abstract: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫semiregular for some integers ▫$m \ge 1$▫, ▫$n \ge 2▫$, and where ▫$\sigma$▫ normalizes ▫$\rho$▫, cyclically permuting the orbits of ▫$\rho$▫ in such a way that ▫$\sigma^m$▫ has at least one fixed vertex. A halfarctransitive graph is a vertex and edge but not arctransitive graph. In this article quartic halfarctransitive metacirculants are explored and their connection to the so called tightly attached quartic halfarctransitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic halfarctransitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed. Found in: ključnih besedah Summary of found: ...Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group... Keywords: mathematics, graph theory, metacirculant graph, halfarctransitive graph, tightly attached, automorphism group Published: 15.10.2013; Views: 1733; Downloads: 68 Full text (0,00 KB) 
19. Consistent Cycles in 1/2ArcTransitive GraphsMarko Boben, Štefko Miklavič, Primož Potočnik, 2009, original scientific article Found in: ključnih besedah Summary of found: ...mathematics, graph theory, 1/2arctransitivity, consistent cycle, ... Keywords: mathematics, graph theory, 1/2arctransitivity, consistent cycle Published: 15.10.2013; Views: 2214; Downloads: 7 Full text (0,00 KB) This document has more files! More...

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