  We consider the notions of sum graph and of relaxed sum graph over a magma,
give several examples and results of these families of graphs over some natural
magmas. We classify the cycles that are sum graphs for the magma of the subsets
of a set with the operation of union, determine the abelian groups that provide
a sum labelling of $C_4$, and show that $C_{4\ell}$ is a sum graph over the
abelian group $\mathbb{Z}_f\times\mathbb{Z}_f$, where $f=f_{2\ell}$ is the
corresponding Fibonacci number. For integral sum graphs, we give a linear upper
bound for the radius of matchings, improving Harary's labelling for this family
of graphs, and give the exact radius for the family of totally disconnected
graphs.
  We found integer labellings for the 4D-cube, giving a negative answer to a
question of Melnikov and Pyatikin, actually showing that the 4D-cube has
infinitely many primitive labellings. We have also obtained some new results on
mod sum graphs and relaxed sum graphs. Finally, we show that the direct product
operation is closed for strong integral sum graphs.
