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Vertex-transitive graphs and their arc-types
Marston D. E. Conder, Tomaž Pisanski, Arjana Žitnik, 2017, izvirni znanstveni članek

Opis: Let ▫$X$▫ be a finite vertex-transitive graph of valency ▫$d$▫, and let ▫$A$▫ be the full automorphism group of ▫$X$▫. Then the arc-type of ▫$X$▫ is defined in terms of the sizes of the orbits of the stabiliser ▫$A_v$▫ of a given vertex ▫$v$▫ on the set of arcs incident with ▫$v$▫. Such an orbit is said to be self-paired if it is contained in an orbit ▫$\Delta$▫ of ▫$A$▫ on the set of all arcs of v$X$▫ such that v$\Delta$▫ is closed under arc-reversal. The arc-type of ▫$X$▫ is then the partition of ▫$d$▫ as the sum ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, where ▫$n_1, n_2, \dots, n_t$▫ are the sizes of the self-paired orbits, and ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ▫$1+1$▫ and ▫$(1+1)$▫, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type.
Ključne besede: symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, cartesian product, covering graph
Objavljeno v RUP: 03.01.2022; Ogledov: 737; Prenosov: 17
.pdf Celotno besedilo (475,17 KB)

3.
On colour-preserving automorphisms of Cayley graphs
Ademir Hujdurović, Klavdija Kutnar, Dave Witte Morris, Joy Morris, 2016, izvirni znanstveni članek

Opis: We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups ▫$G$▫, such that every such automorphism of every connected Cayley graph on ▫$G$▫ has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
Ključne besede: Cayley graph, automorphism, colour-preserving, colour-permuting
Objavljeno v RUP: 03.01.2022; Ogledov: 652; Prenosov: 16
.pdf Celotno besedilo (412,93 KB)

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Existence of non-Cayley Haar graphs
Yan-Quan Feng, István Kovács, Jie Wang, Da-Wei Yang, 2020, izvirni znanstveni članek

Ključne besede: graph, Cayley graph, Haar graph
Objavljeno v RUP: 17.06.2020; Ogledov: 1167; Prenosov: 98
URL Povezava na celotno besedilo

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On groups all of whose Haar graphs are Cayley graphs
Yan-Quan Feng, István Kovács, Da-Wei Yang, 2019, izvirni znanstveni članek

Ključne besede: graph automorphism, Cayley graph, Haar graph
Objavljeno v RUP: 28.06.2019; Ogledov: 1787; Prenosov: 340
URL Povezava na celotno besedilo
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Graphs vertex-partitionable into strong cliques
Ademir Hujdurović, 2019, izvirni znanstveni članek

Ključne besede: canonical double cover, Cayley graph, generalized Cayley graph
Objavljeno v RUP: 28.06.2019; Ogledov: 1643; Prenosov: 233
URL Povezava na celotno besedilo

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