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Recently, perfect state transfer (PST for short) on graphs has attracted great attention due to their applications in quantum information processing and quantum computations. Many constructions and results have been established through various graphs. However, most of the graphs previously investigated are abelian Cayley graphs. Necessary and sufficient conditions for Cayley graphs over dihedral groups having perfect state transfer were studied recently. The key idea in that paper is the assumption of the normality of the connection set. In those cases, viewed as an element in a group algebra, the connection set is in the center of the group algebra, which makes the situations just like in the abelian case. In this paper, we study the non-normal case. In this case, the discussion becomes more complicated. Using the representations of the dihedral group $D_n$, we show that ${\rm Cay}(D_n,S)$ cannot have PST if $n$ is odd. For even integers $n$, it is proved that if ${\rm Cay}(D_n,S)$ has PST, then $S$ is normal.

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Arc-Transitive Graphs of Square-free Order with Valency 11

Algebra Colloquium ◽

10.1142/s100538672100050x ◽

2021 ◽

Vol 28 (04) ◽

pp. 645-654

Author(s):

Guang Li ◽

Bo Ling ◽

Zaiping Lu

Keyword(s):

Simple Group ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Cayley Graphs ◽

Complete List ◽

Dihedral Groups ◽

Transitive Graphs ◽

Free Order ◽

Normal Cayley Graphs

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

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One–regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

Acta Mathematica Sinica English Series ◽

10.1007/s10114-005-0752-9 ◽

2006 ◽

Vol 22 (5) ◽

pp. 1305-1320 ◽

Cited By ~ 21

Author(s):

Jin Ho Kwak ◽

Ju Mok Oh

Keyword(s):

Cayley Graphs ◽

Dihedral Groups ◽

Normal Cayley Graphs

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One-regular cubic graphs of order a small number times a prime or a prime square

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700009903 ◽

2004 ◽

Vol 76 (3) ◽

pp. 345-356 ◽

Cited By ~ 27

Author(s):

Yan-Quan Feng ◽

Jin Ho Kwak

Keyword(s):

Automorphism Group ◽

Cayley Graphs ◽

Cubic Graphs ◽

Cubic Graph ◽

Dihedral Groups ◽

Fixed Order

AbstractA graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2p or 2p2 where p is a prime if and only if 3 is a divisor of p – 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4p or 4p2.

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