Geodesic
Integral Cayley graphs over Abelian groups
The electronic journal of combinatorics, Tome 17 (2010)
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Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.
DOI : 10.37236/353
Classification : 05C25, 05C50 
@article{10_37236_353,
     author = {Walter Klotz and Torsten Sander},
     title = {Integral {Cayley} graphs over {Abelian} groups},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/353},
     zbl = {1189.05074},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/353/}
}
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%J The electronic journal of combinatorics
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Walter Klotz; Torsten Sander. Integral Cayley graphs over Abelian groups. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/353

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