2013
DOI: 10.37236/2742
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Integral Cayley Multigraphs over Abelian and Hamiltonian Groups

Abstract: It is shown that a Cayley multigraph over a group G with generating multiset S is integral (i.e., all of its eigenvalues are integers) if S lies in the integral cone over the boolean algebra generated by the normal subgroups of G. The converse holds in the case when G is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be i… Show more

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Cited by 15 publications
(12 citation statements)
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“…It should be mentioned that the subsets S of Q 8 ×C d p (p a prime) for which the Cayley graph Cay(Q 8 ×C d p , S) is integral are studied in the last section of [7]. We cannot derive the above result from discussions in [7].…”
Section: P:=characteristicpolynomial(a); Fp:=factors(p);mentioning
confidence: 98%
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“…It should be mentioned that the subsets S of Q 8 ×C d p (p a prime) for which the Cayley graph Cay(Q 8 ×C d p , S) is integral are studied in the last section of [7]. We cannot derive the above result from discussions in [7].…”
Section: P:=characteristicpolynomial(a); Fp:=factors(p);mentioning
confidence: 98%
“…The following result has its own interest and we do not use it in the sequel but we would like to state it here! Proposition 2.4 (See Theorem 2 of [7]). Let G be a finite group and S be a member of the boolean algebra generated by the normal subgroups of G. Then the Cayley graph Cay(G, S) is integral.…”
Section: Preliminariesmentioning
confidence: 99%
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“…We were motivated by the natural question, common in representation theory, of whether results known for real reflection groups can be extended to complex reflection groups, often with the goal of better understanding these more general groups. However, questions related to integral adjacency spectra date back to [19] and have been explored extensively for trees and other graphs with special vertex properties (see [5]) as well as for Cayley graphs of abelian groups-in particular for circulant graphs, where the group is cyclic [8,37,2]-and for nonabelian groups with nicely behaved connection sets in [13,18]. Two significant classification results regarding groups that have Cayley graphs with integral adjacency spectrum appear in [1].…”
Section: Introductionmentioning
confidence: 99%
“…In the same year, Klotz and Sander [13] proved that if the Cayley graph Cay(Γ, S) over abelian group Γ is integral, then S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ, and its converse proved by Alperin and Peterson [3]. In 2013, DeVos et al [8] gave a sufficient condition for the integrality of Cayley multigraphs and proved the necessary part for abelian groups, which in turn, is an alternative, character-theoretic proof of the characterization of Bridges and Mena [6]. In 2014, Cheng et al [14] proved that normal Cayley graphs (its generating set S is closed under conjugation) of symmetric groups are integral.…”
mentioning
confidence: 99%