Isomorphisms of Cayley multigraphs of degree 4 on finite abelian groups

Abstract The correspondence between finite abelian groups and their Cayley graphs is studied in the case of degree 4. We show that, but for a simple family of exceptions, the graph is sufficient to determine up to isomorphism the group and the set giving the edges. Ada´m's conjecture about isomorphisms of circulant graphs with degree 4 is thus proven.