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41.
On Hamiltonicity of circulant digraphs of outdegree three
Štefko Miklavič, Primož Šparl, 2009, original scientific article

Abstract: This paper deals with Hamiltonicity of connected loopless circulant digraphs of outdegree three with connection set of the form ▫$\{a,ka,c\}$▫, where ▫$k$▫ is an integer. In particular, we prove that if ▫$k=-1$▫ or ▫$k=2$▫ such a circulant digraph is Hamiltonian if and only if it is not isomorphic to the circulant digraph on 12 vertices with connection set ▫$\{3,6,4\}$▫.
Found in: ključnih besedah
Summary of found: ... graph theory, circulant digraph, Hamilton cycle...
Keywords: graph theory, circulant digraph, Hamilton cycle
Published: 15.10.2013; Views: 1435; Downloads: 59
URL Full text (0,00 KB)

42.
Classification of half-arc-transitive graphs of order 4p
Klavdija Kutnar, Dragan Marušič, Primož Šparl, Ru-Ji Wang, Ming-Yao Xu, 2013, original scientific article

Found in: ključnih besedah
Summary of found: ... graph, ...
Keywords: graph
Published: 15.10.2013; Views: 1270; Downloads: 10
URL Full text (0,00 KB)

43.
Classification of edge-transitive rose window graphs
István Kovács, Klavdija Kutnar, Dragan Marušič, 2010, original scientific article

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 ▫$\{\{x_i, x_{i+1}\} \vert i \in {\mathbb Z}_n\} \cup \{\{y_i, y_{i+r}\} \vert i \in {\mathbb Z}_n\} \cup \{\{x_i, y_i\} \vert i \in {\mathbb Z}_n\} \cup \{\{x_{i+a}, y_i\} \vert i \in {\mathbb Z}_n\}$▫. In this article a complete classification of edge-transitive rose window graphs is given, thus solving one of three open problems about these graphs posed by Steve Wilson in 2001.
Found in: ključnih besedah
Summary of found: ...a$▫, ▫$r \le n-1$▫, the rose window graph ▫$R_n(a,r)$▫ is a quartic graph with vertex...
Keywords: group, graph, rose window, vertex-transitive, edge-transitive, arc-transitive
Published: 15.10.2013; Views: 1441; Downloads: 57
URL Full text (0,00 KB)

44.
On cubic non-Cayley vertex-transitive graphs
Klavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific article

Found in: ključnih besedah
Summary of found: ...vertex-transitive graph, non-Cayley graph, automorphism group, ...
Keywords: vertex-transitive graph, non-Cayley graph, automorphism group
Published: 15.10.2013; Views: 1434; Downloads: 72
URL Full text (0,00 KB)

45.
An introduction to graph theory
Dragan Marušič, 2006, other educational material

Found in: ključnih besedah
Keywords: graph theory
Published: 15.10.2013; Views: 1779; Downloads: 68
URL Full text (0,00 KB)

46.
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.
Found in: ključnih besedah
Summary of found: ...Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with vertex set ▫$X$▫, diameter ▫$d \ge...
Keywords: mathematics, graph theory, distance-regular graphs, ▫$Q$▫-polynominal property, Bose-Mesner algebra, balanced set characterization of the Q-polynominal property
Published: 15.10.2013; Views: 1751; Downloads: 12
URL Full text (0,00 KB)

47.
Adjacency preservers, symmetric matrices, and cores
Marko Orel, 2012, original scientific article

Abstract: It is shown that the graph ▫$\Gamma_n$▫ that has the set of all ▫$n \times n$▫ symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if ▫$n \ge 3$▫. Eigenvalues of the graph ▫$\Gamma_n$▫ are calculated as well.
Found in: ključnih besedah
Summary of found: ...It is shown that the graph ▫$\Gamma_n$▫ that has the set of all...
Keywords: adjacency preserver, symmetric matrix, finite field, eigenvalue of a graph, coloring, quadratic form
Published: 15.10.2013; Views: 1625; Downloads: 85
URL Full text (0,00 KB)

48.
Group irregularity strength of connected graphs
Marcin Anholcer, Sylwia Cichacz, Martin Milanič, 2013, original scientific article

Found in: ključnih besedah
Summary of found: ...irregularity strength, graph labelling, Abelian group, ...
Keywords: irregularity strength, graph labelling, Abelian group
Published: 15.10.2013; Views: 975; Downloads: 58
URL Full text (0,00 KB)

49.
A spectral proof of the uniqueness of a strongly regular graph with parameters (81, 20, 1, 6)
Dragan Stevanović, Marko Milošević, 2009, original scientific article

Abstract: We give a new proof that there exists a unique strongly regular graph with parameters (81, 20, 1, 6). Unlike the finite geometry approach used by Brouwerand haemers, we use linear algebra and spectral graph theory concepts, namely the technique of star complements, in our proof.
Found in: ključnih besedah
Keywords: graph theory
Published: 15.10.2013; Views: 1450; Downloads: 10
URL Full text (0,00 KB)

50.
Hamilton cycles in (2, odd, 3)-Cayley graphs
Henry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific article

Abstract: In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.
Found in: ključnih besedah
Summary of found: ...Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of...
Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group
Published: 15.10.2013; Views: 1356; Downloads: 69
URL Full text (0,00 KB)

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