Sum graphs over all the integers

AbstractWe introduced the sum graph of a set S of positive integers as the graph G+(S) having S as its node set, with two nodes adjacent whenever their sum is in S. Now we study sum graphs over all the integers so that S may contain positive or negative integers on zero. A graph so obtained is called an integral sum graph. The sum number of a given graph G was defined as the smallest number of isolated nodes which when added to G result in a sum graph. The integral sum number of G is analogous. We see that all paths and all matchings are integral sum graphs. We find the integral sum number of the small graphs and offer several intriguing unsolved problems