1. Vertex-transitive expansions of (1, 3)-treesMarko Lovrečič Saražin, Dragan Marušič, 2010, published scientific conference contribution Abstract: A nonidentity automorphism of a graph is said to be semiregular if all of its orbits are of the same length. Given a graph ▫$X$▫ with a semiregular automorphism ▫$\gamma$▫, the quotient of ▫$X$▫ relative to ▫$\gamma$▫ is the multigraph ▫$X/\gamma$▫ whose vertices are the orbits of ▫$\gamma$▫ and two vertices are adjacent by an edge with multiplicity ▫$r$▫ if every vertex of one orbit is adjacent to ▫$r$▫ vertices of the other orbit. We say that ▫$X$▫ is an expansion of ▫$X/\gamma$▫. In [J.D. Horton, I.Z. Bouwer, Symmetric ▫$Y$▫-graphs and ▫$H$▫-graphs, J. Combin. Theory Ser. B 53 (1991) 114-129], Hortonand Bouwer considered a restricted sort of expansions (which we will call :strong" in this paper) where every leaf of ▫$X/\gamma$▫ expands to a single cycle in ▫$X$▫. They determined all cubic arc-transitive strong expansions of simple ▫$(1,3)$▫-trees, that is, trees with all of their vertice shaving valency 1 or 3, thus extending the classical result of Frucht, Graver and Watkins (see [R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211-218]) about arc-transitive strong expansions of ▫$K_2$▫ (also known as the generalized Petersen graphs). In this paper another step is taken further by considering the possible structure of cubic vertex-transitive expansions of general ▫$(1,3)$▫-multitrees (where vertices with double edges are also allowed); thus the restriction on every leaf to be expanded to a single cycle is dropped. Found in: ključnih besedah Summary of found: Zadetek v naslovu Keywords: graph, tree, cubic, vertex-transitive, arc-transitive, expansion Published: 15.10.2013; Views: 2265; Downloads: 56 Full text (0,00 KB) |
2. Hamilton paths in vertex-transitive graphs of order 10pKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific article Abstract: It is shown that every connected vertex-transitive graph of order ▫$10p$▫, ▫$p \ne 7$▫ a prime, which is not isomorphic to a quasiprimitive graph arising from the action of PSL▫$(2,k)$▫ on cosets of ▫$\mathbb{Z}_k \times \mathbb{Z}_{(k-1)/10}$▫, contains a Hamilton path. Found in: ključnih besedah Summary of found: Zadetek v naslovu Keywords: graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Published: 15.10.2013; Views: 2022; Downloads: 14 Full text (0,00 KB) |
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4. On prime-valent symmetric bicirculants and Cayley snarksAdemir Hujdurović, Klavdija Kutnar, Dragan Marušič, 2013, published scientific conference contribution Found in: ključnih besedah Summary of found: ...graph, Cayley graph, arc-transitive, snark, semiregular automorphism, bicirculant, ... Keywords: graph, Cayley graph, arc-transitive, snark, semiregular automorphism, bicirculant Published: 15.10.2013; Views: 1836; Downloads: 91 Full text (0,00 KB) |
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6. Distance-balanced graphs: Symmetry conditionsKlavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2006, original scientific article Abstract: A graph ▫$X$▫ is said to be distance-balanced 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 distance-balanced 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) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced 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 edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced 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: ...... Keywords: graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graph Published: 15.10.2013; Views: 2197; Downloads: 61 Full text (0,00 KB) |
7. Classification of 2-arc-transitive dihedrantsShao Fei Du, Aleksander Malnič, Dragan Marušič, 2008, original scientific article Abstract: A complete classification of 2-arc-transitive 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 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. 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 ▫$\frac{q-1}{2}$▫ and ▫$q-1$▫, 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 ▫$i-j = \theta^h$▫, where ▫$\theta$▫ generates the multiplicative group ▫${\mathbb{F}}_q^\ast$▫ of the Galois field ▫${\mathbb{F}}_q$▫. Found in: ključnih besedah Summary of found: Zadetek v naslovu Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph Published: 15.10.2013; Views: 1850; Downloads: 59 Full text (0,00 KB) |
8. Rose window graphs underlying rotary mapsIstván Kovács, Klavdija Kutnar, János Ruff, 2010, published scientific conference contribution Abstract: Given natural numbers ▫$n \ge 3$▫ and ▫$1 \le a$▫, ▫$r \le n-1$▫, 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), 7-19. 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: ...... Keywords: graph theory, rotary map, edge-transitive graph, covering graph, voltage graph Published: 15.10.2013; Views: 1853; Downloads: 60 Full text (0,00 KB) |
9. On quartic half-arc-transitive 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 half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive 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: Zadetek v naslovu Keywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group Published: 15.10.2013; Views: 1934; Downloads: 78 Full text (0,00 KB) |
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