Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

Klotz, Walter; Sander, Torsten

doi:10.37236/2369

Cited by 17 publications

(17 citation statements)

References 13 publications

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“…In this article we extend the result of Klotz and T. Sander [7] in the case where G is a cyclic group. It turns out that in this situation the above phenomenon can be explained much more explicitly.…”

Section: Introductionsupporting

confidence: 56%

“…The theorem of Klotz and T. Sander [7] now reads: Let I and J be two ideals in the residue class ring Z n . Then I ⋆ + J ⋆ is the (disjoint) union of atoms in Z n , i.e.…”

Section: Terminology and The Reduction Lemmamentioning

confidence: 99%

“…In this case it refers to the Boolean algebra generated by the subgroups of G. It is not difficult to see that every element of this Boolean algebra is a disjoint union of atoms. Recently Klotz and the second author proved the following result in [7]: Let G be a finite abelian group with a, b ∈ G. Then atom(a) + atom(b) is the (disjoint) union of atoms of G.…”

Section: Introductionmentioning

confidence: 99%

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Adding generators in cyclic groups

Sander

2013

Journal of Number Theory

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For a cyclic group $a$, define the atom of $a$ as the set of all elements generating $a$. Given any two elements $a,b$ of a finite cyclic group $G$, we study the sumset of the atom of $a$ and the atom of $b$. It is known that such a sumset is a disjoint union of atoms. The goal of this paper is to offer a deeper understanding of this phenomenon, by determining which atoms make up the sum of two given atoms and by computing the exact number of representations of each element of the sumset

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Section: Introductionsupporting

confidence: 56%

“…The theorem of Klotz and T. Sander [7] now reads: Let I and J be two ideals in the residue class ring Z n . Then I ⋆ + J ⋆ is the (disjoint) union of atoms in Z n , i.e.…”

Section: Terminology and The Reduction Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adding generators in cyclic groups

Sander

2013

Journal of Number Theory

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“…If w h = 0, then t = w h + 1 = 1, which is impossible. If w h = 1, then t = 2, and p i = 3 by (12). Next assume that t − w h > 1.…”

Section: Integral Cayley Graphs Over D Pmentioning

confidence: 99%

Integral Cayley graphs over dihedral groups

Huang

2017

J Algebr Comb

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In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group D n = a, b | a n = b 2 = 1, bab = a −1 . Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over D n in terms of the Boolean algebra of a , from which we find infinite classes of integral Cayley graphs over D n . In particular, we completely determine all integral Cayley graphs over the dihedral group D p for a prime p.

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“…Two significant classification results regarding groups that have Cayley graphs with integral adjacency spectrum appear in [1]. There are also contributions relating adjacency and distance spectra of Cayley graphs of cyclic groups [22], and more generally abelian groups, even for a broad class of connection sets [23].…”

Section: Introductionmentioning

confidence: 99%

Spectra of Cayley graphs of complex reflection groups

Foster-Greenwood

Kriloff

2015

J Algebr Comb

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ABSTRACT. Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.

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Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

Abstract: It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

Cited by 17 publications

References 13 publications

Adding generators in cyclic groups

Adding generators in cyclic groups

Integral Cayley graphs over dihedral groups

Spectra of Cayley graphs of complex reflection groups

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