Proof of the conjecture about edge - magic graph

Let G be a graph with p vertices and q edges. Assume the vertices and edges of G are labelled by 1, 2, 3, ..., p + q such that each label is used exactly once. We define the valence of an edge to be the sum of the label of e plus the two labels of the vertices incident with e. If a labelling of G is possible such that the valence for e is constant, we call the graph G is edge - magic. A caterpillar T is a tree in which the removal of all end - nodes results in a path (called the spine of T). For examples, the paths and the stars are caterpillars. In this paper, we prove caterpillars are edge - magic. So we prove that the tree with at most 8 nodes is edge - magic.