Integral Cayley multigraphs over abelian and Hamiltonian groups
Summary: It is shown that a Cayley multigraph over a group \(G\) with generating multiset \(S\) is integral (i.e., all of its eigenvalues are integers) if \(S\) lies in the integral cone over the boolean algebra generated by the normal subgroups of \(G\). The converse holds in the case when \(G\) is abelian. This in particular gives an alternative, character theoretic proof of a theorem of \textit{W. G. Bridges} and \textit{R. A. Mena} [J. Comb. Theory, Ser. A 32, 264--280 (1982; Zbl 0485.05040)]. We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set.
scientific article; zbMATH DE number 6330271