Integral Cayley Multigraphs over Abelian and Hamiltonian Groups

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 Bridges and Mena (1982). We extend this result by providing 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.