Spectra of Cayley graphs of complex reflection groups

Foster-Greenwood, Briana; Kriloff, Cathy

doi:10.1007/s10801-015-0652-8

Cited by 13 publications

(9 citation statements)

References 36 publications

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“…Using this result, the authors gave there necessary and sufficient conditions on S for Cay(G, S) to be distance integral in the case where G is a generalized dihedral group, and they obtained a similar result for dicyclic groups in [HL21a]. See also [Ren11,FK16] for earlier works over specific finite groups, with a geometric flavour.…”

Section: Introductionmentioning

confidence: 92%

On Cayley graphs over generalized dicyclic groups

Behajaina,

Legrand

2022

Preprint

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Recently, several works by a number of authors have studied integrality, distance integrality, and distance powers of Cayley graphs over some finite groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is to generalize and/or to give analogues of these results for generalized dicyclic groups. For example, we give a necessary and sufficient condition for a Cayley graph over a generalized dicyclic group to be integral (i.e., all eigenvalues of its adjacency matrix are in Z). We also obtain sufficient conditions for the integrality of all distance powers of a Cayley graph over a given generalized dicyclic group. These results extend works on dicyclic groups by Cheng-Feng-Huang and Cheng-Feng-Liu-Lu-Stevanovic, respectively.

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Section: Introductionmentioning

confidence: 92%

On Cayley graphs over generalized dicyclic groups

Behajaina,

Legrand

2022

Preprint

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“…Theorem 2.2. [11] Let α : Γ → C be a class function and Irr(Γ) = {χ 1 , ..., χ h }. Then the spectrum of the Cayley color digraph Cay(Γ, α) can be arranged as…”

Section: Preliminariesmentioning

confidence: 99%

HS-integral and Eisenstein integral normal mixed Cayley graphs

Kadyan¹,

Bhattacharjya²

2022

Preprint

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“…In the special case when α : Γ → {0, 1, i, −i} with α(s) = α(s −1 ) and the set Theorem 2.1. [12] Let Γ be a finite group, Irr(Γ) = {χ 1 , ..., χ h } and α : Γ → C be a class function.…”

Section: Normal Mixed Cayley Graph and Group Charactersmentioning

confidence: 99%

H-integral and Gaussian integral normal mixed Cayley graphs

Kadyan¹

2021

Preprint

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A mixed graph is called integral if all the eigenvalues of its Hermitian adjacency matrix are integers.A mixed Cayley graph Cay(Γ, S) is called normal if S is the union of some conjugacy classes of a finite group Γ. In 2014, Godsil and Spiga characterized integral normal Cayley graphs. We give similar characterization for the integrality of a normal mixed Cayley graph Cay(Γ, S) in terms of S.Xu and Meng (2011) and Li (2013) characterized the set S ⊆ Z n for which the eigenvalues k∈S w jk n of the circulant digraph Cay(Z n , S) are Gaussian integers for all j = 1, ..., h. Here the adjacency matrix of Cay(Z n , S) is considered to be the n × n matrix [a ij ], where a ij = 1 if (i, j) is an arc of Cay(Z n , S), and 0 otherwise.Let {χ 1 , . . . , χ h } be the set of the irreducible characters of Γ. We prove that 1 χj (1) s∈S χ j (s) is aGaussian integer for all j = 1, ..., h if and only if the normal mixed Cayley graph Cay(Γ, S) is integral.As a corollary to this, we get an alternative and easy proof of the characterization, as obtained by Xu, Meng and Li, of the set S ⊆ Z n for which the circulant digraph Cay(Z n , S) is Gaussian integral.

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Spectra of Cayley graphs of complex reflection groups

Cited by 13 publications

References 36 publications

On Cayley graphs over generalized dicyclic groups

On Cayley graphs over generalized dicyclic groups

HS-integral and Eisenstein integral normal mixed Cayley graphs

H-integral and Gaussian integral normal mixed Cayley graphs

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