Integral Cayley graphs over dicyclic group

How to classify all the integral graphs is a challenging work as suggested by Harary and Schwenk. In this paper, we focus on one non-abelian group—the dicyclic group T4n=〈a,b|a2n=1,an=b2,b−1ab=a−1〉, and consider its corresponding Cayley graphs. With the help of its character table, we first obtain a...

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Bibliographic Details
Published in:Linear algebra and its applications 2019-04, Vol.566, p.121-137
Main Authors: Cheng, Tao, Feng, Lihua, Huang, Hualin
Format: Article
Language:English
Subjects:
Boolean algebra
Character
Dicyclic groups
Eigenvalue
Graphs
Group theory
Integral Cayley graph
Integrals
Linear algebra
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Summary:How to classify all the integral graphs is a challenging work as suggested by Harary and Schwenk. In this paper, we focus on one non-abelian group—the dicyclic group T4n=〈a,b|a2n=1,an=b2,b−1ab=a−1〉, and consider its corresponding Cayley graphs. With the help of its character table, we first obtain a necessary and sufficient condition for the integrality of Cayley graphs over T4n. Then we obtain several simple sufficient conditions for the integrality of Cayley graphs over T4n in terms of the Boolean algebra of 〈a〉. As a byproduct, we determine a few infinite families of connected integral Cayley graphs over T4n. At last, for a prime p, we completely determine all integral Cayley graphs over the dicyclic group T4p.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2019.01.002