A magic graph is a graph whose edges are labelled by the first q positive integers, where q is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. The name "magic" sometimes means that the integers are any positive integers; then the graph and the labelling using the first q positive integers are called supermagic.
A graph is vertex-magic if its vertices can be labelled so that the sum on any edge is the same. It is total magic if its edges and vertices can be labelled so that the vertex label plus the sum of labels on edges incident with that vertex is a constant.
There are a great many variations on the concept of magic labelling of a graph. There is much variation in terminology as well. The definitions here are perhaps the most common.
Comprehensive references for magic labellings and magic graphs are Gallian (1998), Wallis (2001), and Marr and Wallis (2013).