On integral Cayley sum graphs (Q2360579)

Let \(G\) be a finite abelian group, and let \( S\) be any subset (possibly a multiset) of \(G\). The Cayley sum graph \({\text{{Cay}}^+}(G,S)\) is the graph with vertex set \(G\) and edge set \(\{\{g,h\}:g,h\in G;\;g+h\in S\}\). A graph is integral if all eigenvalues of its adjacency matrix are integers. A group \(G\) is Cayley sum integral if \({\text{{Cay}}^+}(G,S)\) is integral for all \(S\subseteq G\). The first main result is that a finite abelian group \(G\) is Cayley sum integral if and only if either \(G\cong\mathbb{Z}_2^n\) for some \(n\) or \(G\cong\mathbb{Z}_3\). The second main result is that there exist exactly six simple, connected, cubic integral Cayley sum graphs.