Integral sum graphs from identification

The idea of integral sum graphs was introduced by Harary (1994). A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u) + f(v) = f(w) for some node w in G. A tree is said to be a generalized star if it can be obtained from a star by extending each edge to a path. A node of a tree T is said to be a fork of T if its degree is not equal to two. In this paper, we first introduce some methods of identification on constructing new connected integral sum graphs from given integral sum graphs. Applying the methods of identification, we then prove that the generalized stars and the trees with all forks at least distance 4 apart are integral sum graphs.