On the rational spectra of graphs with abelian singer groups

Abstract

Let 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.