On the rational spectra of graphs with abelian singer groups

AbstractLet G be a finite abelian group. We investigate those graphs G admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra P(G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in P(G) for various G's, we relate subschemes of a related association scheme, subalgebras of P(G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors