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A method to determine algebraically integral Cayley digraphs on finite abelian group. (English) Zbl 1477.05084

The author considers the problem of characterizing Cayley digraphs on any given finite abelian group, such that their eigenvalues are algebraic integers over a given number field. The main result is that a Cayley digraph generated using a subset \(S\) of a finite abelian group \(G\) of order \(n\), is integral over a number field \(K\) if and only if \(S\) is a union of orbits under the natural action of \(\mathrm{Gal}(\mathbb{Q}(\zeta_n)/K)\) i.e. the Galois group of the \(n\)-th cyclotomic extension of \(\mathbb{Q}\) over \(K\). As a corollary, the author also obtains an expression for the number of such Cayley digraphs: \(2^{r(G,K)}\), where \(r(G,K)\) is the number of non-zero orbits of \(G\) under the above-mentioned group action. These results also apply to the case of Gaussian integral Cayley digraphs on a finite abelian group, where a Cayley digraph is Gaussian integral if it is integral over the Gaussian field \(\mathbb{Q}(i)\).
The results and the proofs extend the earlier work of the author [Discrete Math. 313, No. 6, 821–823 (2013; Zbl 1260.05068)], which answered the corresponding question for the case of circulant digraphs – i.e. Cayley digraphs over cyclic groups. This had solved a conjecture of Y. Xu and J. Meng [ibid. 311, No. 1, 45–50 (2011; Zbl 1225.05121)]. To extend the proof for the circulant case to Cayley digraphs over arbitrary finite abelian groups, the idea is to decompose the finite abelian group into a direct sum of cyclic groups using the structure theorem of abelian groups.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20K99 Abelian groups
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References:

[1] R. C. Alperin and B. L. Peterson,Integral sets and Cayley graphs of finite groups, Electron. J. Combin.19(2012), #P44. · Zbl 1243.05143
[2] K. Babai,Spectra of Cayley graphs, J. Comb. Theory Ser. B27(1979), 180-189. · Zbl 0338.05110
[3] K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic, and D. Stevanovic,A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.13(2002), 42-65. · Zbl 1051.05057
[4] N. Biggs,Algebraic graph theory, North-Holland, Amsterdam, 1985. · Zbl 0797.05032
[5] W. G. Bridges and R. A. Mena,Rational G-matrices with rational eigenvalues, J. Comb. Theory Ser. A32(1982), no. 2, 264-280. · Zbl 0485.05040
[6] F. Esser and F. Harary,Digraphs with real and Gaussian spectra, Discrete Appl. Math. 2(1980), 113-124. · Zbl 0435.05027
[7] C. Godsil and G. Royle,Algebraic graph theory, Springer-Verlag, New York, 2001. · Zbl 0968.05002
[8] F. Harary and A. Schwenk,Which graphs have integral spectra, Graphs Combin. (R. Bari and F. Harary, eds.), Springer-Verlag, Berlin, 1974, pp. 45-51.
[9] X. D. Hou,On the G-matrices with entries and eigenvalues inQ(i), Graphs Combin. 8(1992), 53-64.
[10] S. Lang,Algebra, 3rd ed., Springer-Verlag, New York, 2002. · Zbl 0984.00001
[11] F. Li,Circulant digraphs integral over number fields, Discrete Math.313(2013), 821- 823. · Zbl 1260.05068
[12] J. Sander and T. Sander,The exact maxamal energy of integral circulant graphs with prime power order, Contrib. Discrete Math.8(2013), no. 2, 19-40. · Zbl 1317.05114
[13] J. P. Serre,Linear representations of finite groups, Springer-Verlag, New York, 1977. · Zbl 0355.20006
[14] W. So,Integral circulant graphs, Discrete Math.306(2005), 153-158. · Zbl 1084.05045
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