1. Hamiltonicity of vertextransitive graphs of order 4pKlavdija Kutnar, Dragan Marušič, 2008, original scientific article Abstract: It is shown that every connected vertextransitive graph of order ▫$4p$▫, where ▫$p$▫ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path. Found in: ključnih besedah Summary of found: ...It is shown that every connected vertextransitive graph of order ▫$4p$▫, where ▫$p$▫ is a... ...graph theory, vertextransitive graphs, Hamilton cycle, automorphism group... Keywords: graph theory, vertextransitive graphs, Hamilton cycle, automorphism group Published: 15.10.2013; Views: 1626; Downloads: 18 Full text (0,00 KB) 
2. Isomorphism checking of IgraphsBoris Horvat, Tomaž Pisanski, Arjana Žitnik, 2012, original scientific article Abstract: We consider the class of ▫$I$▫graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two ▫$I$▫graphs ▫$I(n, j, k)$▫ and ▫$I(n, j_1, k_1)$▫ are isomorphic if and only if there exists an integer ▫$a$▫ relatively prime to $n$ such that either ▫$\{j_1, k_1\} = \{aj \mod n, \; ak \mod n \}$▫ or ▫$\{j_1, k_1\} = \{aj \mod n, \; ak \mod n\}$▫. This result has an application in the enumeration of nonisomorphic ▫$I$▫graphs and unitdistance representations of generalized Petersen graphs. Found in: ključnih besedah Summary of found: ...mathematics, graph theory, isomorphism, Igraph, generalized Petersen graph... Keywords: mathematics, graph theory, isomorphism, Igraph, generalized Petersen graph Published: 15.10.2013; Views: 1570; Downloads: 73 Full text (0,00 KB) 
3. 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 Summary of found: ...A graph ▫$X$▫ is said to be distancebalanced if... ...graph theory, graph, distancebalanced graphs, vertextransitive, semysimmetric, generalized ... Keywords: graph theory, graph, distancebalanced graphs, vertextransitive, semysimmetric, generalized Petersen graph Published: 15.10.2013; Views: 1900; Downloads: 53 Full text (0,00 KB) 
4. 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... ...graph theory, cubic graphs, symmetric graphs, ▫$s$▫regular graphs, girth,... Keywords: graph theory, cubic graphs, symmetric graphs, ▫$s$▫regular graphs, girth, consistent cycle Published: 15.10.2013; Views: 1829; Downloads: 49 Full text (0,00 KB) 
5. 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)$▫ is a quartic graph with vertex... ...graph theory, rotary map, edgetransitive graph, covering graph, voltage... Keywords: graph theory, rotary map, edgetransitive graph, covering graph, voltage graph Published: 15.10.2013; Views: 1588; Downloads: 49 Full text (0,00 KB) 
6. Qpolynomial distanceregular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0Štefko Miklavič, 2008, original scientific article Abstract: Let ▫$\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... ...mathematics, graph theory, adjacency matrix, distanceregular graph, Terwilliger algebra... Keywords: mathematics, graph theory, adjacency matrix, distanceregular graph, Terwilliger algebra Published: 15.10.2013; Views: 1512; Downloads: 9 Full text (0,00 KB) 
7. 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... ...mathematics, graph theory, metacirculant graph, halfarctransitive graph, tightly attached, automorphism... Keywords: mathematics, graph theory, metacirculant graph, halfarctransitive graph, tightly attached, automorphism group Published: 15.10.2013; Views: 1704; Downloads: 63 Full text (0,00 KB) 
8. 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: 2185; Downloads: 7 Full text (0,00 KB) This document has more files! More...

9. 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: ...positive integer and let ▫${\mathcal{Q}}_D$▫ denote the graph of the ▫$D$▫dimensional hypercube. Let ▫$X$ denote... ...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: 1562; Downloads: 61 Full text (0,00 KB) 
10. 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... ...graph theory, connected graphs, connectivity, distancebalanced graphs, bipartite graphs... Keywords: graph theory, connected graphs, connectivity, distancebalanced graphs, bipartite graphs Published: 15.10.2013; Views: 1348; Downloads: 50 Full text (0,00 KB) 