Sum Graphs from Trees

A graph \(G\) is a sum graph if there is a labeling \(o\) of its vertices with distinct positive integers, so that for any two distinct vertices \(u\) and \(v\), \(uv\) is an edge of \(G\) if and only if \(\sigma(u) +\sigma(v) = \sigma(w)\) for some other vertex \(w\). Every sum graph has at least one isolated vertex (the vertex with the largest label). Harary has conjectured that any tree can be made into a sum graph with the addition of a single isolated vertex. We prove this conjecture.