Three-state quantum walk on the Cayley Graph of the Dihedral Group

For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] the bi-Cayley graph BCay[Formula: see text] of [Formula: see text] with respect to [Formula: see text] is the bipartite graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bi-Cayley graph BCay[Formula: see text] is called a BCI-graph if for any bi-Cayley graph BCay[Formula: see text], [Formula: see text] implies that [Formula: see text] for some [Formula: see text] and [Formula: see text]. A group [Formula: see text] is called a [Formula: see text]-BCI-group if all bi-Cayley graphs of [Formula: see text] with valency at most [Formula: see text] are BCI-graphs. In this paper, we characterize the [Formula: see text]-BCI dihedral groups for [Formula: see text]. Also, we show that the dihedral group [Formula: see text] ([Formula: see text] is prime) is a [Formula: see text]-BCI-group.

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The Energy of Cayley Graphs for a Generating Subset of the Dihedral Groups

MATEMATIKA ◽

10.11113/matematika.v35.n3.1115 ◽

2019 ◽

Vol 35 (3) ◽

Author(s):

Amira Fadina Ahmad Fadzil ◽

Nor Haniza Sarmin ◽

Ahmad Erfanian

Keyword(s):

Finite Group ◽

Cayley Graph ◽

Adjacency Matrix ◽

Dihedral Group ◽

Cayley Graphs ◽

Dihedral Groups ◽

Specific Subset ◽

A Finite Group

Let G be a finite group and S be a subset of G where S does not include the identity of G and is inverse closed. A Cayley graph of a group G with respect to the subset S is a graph where its vertices are the elements of G and two vertices a and b are connected if ab^(−1) is in the subset S. The energy of a Cayley graph is the sum of all absolute values of the eigenvalues of its adjacency matrix. In this paper, we consider a specific subset S = {b, ab, . . . , a^(n−1)b} for dihedral group of order 2n, where n is greater or equal to 3 and find the Cayley graph with respect to the set. We also calculate the eigenvalues and compute the energy of the respected Cayley graphs. Finally, the generalization of the energy of the respected Cayley graphs is found.

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Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2016.0394 ◽

2017 ◽

Vol 375 (2106) ◽

pp. 20160394 ◽

Cited By ~ 1

Author(s):

G. M. D'Ariano ◽

N. Mosco ◽

P. Perinotti ◽

A. Tosini

Keyword(s):

Cayley Graph ◽

Degrees Of Freedom ◽

Transition Amplitude ◽

Quantum Walk ◽

Transition Matrices ◽

Unitary Evolution ◽

Position Representation ◽

Quantum Revolution ◽

The Given ◽

Internal Degrees Of Freedom

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group , which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

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The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

Proyecciones (Antofagasta) ◽

10.22199/issn.0717-6279-4357-4429 ◽

2021 ◽

Vol 40 (6) ◽

pp. 1683-1691

Author(s):

Saba AL-Kaseasbeh ◽

Ahmad Erfanian

Keyword(s):

Cayley Graph ◽

Dihedral Group ◽

Cayley Graphs ◽

Dihedral Groups

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

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Automorphism Groups of Connected Cubic Cayley Graphs of Order 4p

Algebra Colloquium ◽

10.1142/s100538670700034x ◽

2007 ◽

Vol 14 (02) ◽

pp. 351-359 ◽

Cited By ~ 9

Author(s):

Chuixiang Zhou ◽

Yan-Quan Feng

Keyword(s):

Automorphism Group ◽

Cayley Graph ◽

Dihedral Group ◽

Automorphism Groups ◽

Regular Representation ◽

Cayley Graphs ◽

Three Dimensional ◽

Group A ◽

The Right

For a prime p, let D4p be the dihedral group 〈a,b | a2p = b2 = 1, b-1ab = a-1〉 of order 4p, and Cay (G,S) a connected cubic Cayley graph of order 4p. In this paper, it is shown that the automorphism group Aut ( Cay (G,S)) of Cay (G,S) is the semiproduct R(G) ⋊ Aut (G,S), where R(G) is the right regular representation of G and Aut (G,S) = {α ∈ Aut (G) | Sα = S}, except either G = D4p (p ≥ 3), Sβ = {b,ab,apb} for some β ∈ Aut (D4p) and [Formula: see text], or Cay (G,S) is isomorphic to the three-dimensional hypercube Q3[Formula: see text] and G = ℤ4 × ℤ2 or D8.

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