Geodesic
Integral Cayley graphs defined by greatest common divisors
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.
DOI : 10.37236/581
Classification : 05C25, 05C50 
@article{10_37236_581,
     author = {Walter Klotz and Torsten Sander},
     title = {Integral {Cayley} graphs defined by greatest common divisors},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/581},
     zbl = {1217.05105},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/581/}
}
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Walter Klotz; Torsten Sander. Integral Cayley graphs defined by greatest common divisors. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/581

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